aa r X i v : . [ m a t h . F A ] D ec EULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS
GELU POPESCU
Abstract.
In this paper we introduce and study the Euler characteristic (denoted by χ ) associated withalgebraic modules generated by arbitrary elements of certain noncommutative polyballs. We provideseveral asymptotic formulas for χ and prove some of its basic properties. We show that the Eulercharacteristic is a complete unitary invariant for the finite rank Beurling type invariant subspaces of thetensor product of full Fock spaces F ( H n ) ⊗ · · · ⊗ F ( H n k ), and prove that its range coincides withthe interval [0 , ∞ ). We obtain an analogue of Arveson’s version of the Gauss-Bonnet-Chern theoremfrom Riemannian geometry, which connects the curvature to the Euler characteristic. In particular,we prove that if M is an invariant subspace of F ( H n ) ⊗ · · · ⊗ F ( H n k ), n i ≥
2, which is graded(generated by multi-homogeneous polynomials), then the curvature and the Euler characteristic of theorthocomplement of M coincide. Introduction
Arveson ([1], [2], [3]) introduced and studied the curvature invariant and the Euler characteristic forcommuting n -tuples of bounded linear operators T = ( T , . . . , T n ) acting on a Hilbert space H such that T T ∗ + · · · + T n T ∗ n ≤ I and the defect operator ∆ T := ( I − T T ∗ − · · · − T n T ∗ n ) / has finite rank. Basedon certain asymptotic formulas for both the curvature and Euler characteristic, he obtained, as the mainresult of the paper, an operator-theoretic version of the Gauss-Bonnet-Chern formula of Riemannian ge-ometry. More precisely, he proved that for any graded (generated by homogeneous polynomials) invariantsubspace M of a finite direct sum of copies of the symmetric Fock space F s ( H n ) with n generators, thecurvature and the Euler characteristic of the othocomplement of M coincide. This result was generalizedby Fang ([6]) to invariant subspaces generated by arbitrary polynomials. Fang also obtained a version ofthe Gauss-Bonnet-Chern formula for the Hardy space H ( D n ) over the polydisc([8]) and for the Dirichletspace over the unit disc ([7], [25]). The theory of Arveson’s curvature on the symmetric Fock space wassignificantly expanded due to the work by Greene, Richter, and Sundberg [9], Fang [6], and Gleason,Richter, and Sundberg [10].In the noncommutative setting, the author ([18], [19]) and, independently, Kribs [12] defined andstudied the curvature and Euler characteristic for arbitrary elements T with rank ∆ T < ∞ in the unitball [ B ( H ) n ] − := { ( X , . . . , X n ) ∈ B ( H ) n : X X ∗ + · · · + X n X ∗ n ≤ I } and, in particular, for the invariant (resp. coinvariant) subspaces of the full Fock space F ( H n ) with n generators. Some of these results were extended by Muhly and Solel [13] to a class of completely positivemaps on semifinite factors. A noncommutative analogue (for the full Fock space) of the Arveson’s versionof the Gauss-Bonnet-Chern theorem was obtained in [18].In [24], we developed a theory of curvature invariant and multiplicity invariant for tensor productsof full Fock spaces F ( H n ) ⊗ · · · ⊗ F ( H n k ) and also for tensor products of symmetric Fock spaces F s ( H n ) ⊗ · · · ⊗ F s ( H n k ). To prove the existence of the curvature and its basic properties in thesesettings required a new approach based on noncommutative Berezin transforms and multivariable operatortheory on polyballs and varieties (see [17], [21], [22], and [23]), and also certain summability results for Date : July 9, 2014 (revised version).2000
Mathematics Subject Classification.
Primary: 46L52; 32A70; Secondary: 47A13; 47A15.
Key words and phrases.
Noncommutative polyball; Euler characteristic; Curvature invariant; Berezin transform; Fockspace; Creation operators; Invariant subspaces.Research supported in part by an NSF grant. completely positive maps which are trace contractive. In particular, we obtained new proofs for theexistence of the curvature on the full Fock space F ( H n ), the Hardy space H ( D k ) (which correspondsto n = · · · = n k = 1), and the symmetric Fock space F s ( H n ).In the present paper, which is a continuation of [24], we introduce and study the Euler characteristicassociated with the elements of a class of noncommutative polyballs, and obtain an analogue of Arveson’sversion of the Gauss-Bonnet-Chern theorem from Riemannian geometry, which connects the curvatureto the Euler characteristic of some associated algebraic modules.To present our results we need a few preliminaries. Throughout this paper, we denote by B ( H )the algebra of bounded linear operators on a Hilbert space H . Let B ( H ) n × c · · · × c B ( H ) n k be theset of all tuples X := ( X , . . . , X k ) in B ( H ) n × · · · × B ( H ) n k with the property that the entries of X s := ( X s, , . . . , X s,n s ) are commuting with the entries of X t := ( X t, , . . . , X t,n t ) for any s, t ∈ { , . . . , k } , s = t . Note that, for each i ∈ { , . . . , k } , the operators X i, , . . . , X i,n i are not necessarily commuting.Let n := ( n , . . . , n k ) ∈ N k , where n i ∈ N := { , , . . . } and i ∈ { , . . . , k } , and define the regular polyball B n ( H ) := { X = ( X , . . . , X k ) ∈ B ( H ) n × c · · · × c B ( H ) n k : ∆ pX ( I ) ≥ ≤ p ≤ (1 , . . . , } , where the defect mapping ∆ pX : B ( H ) → B ( H ) is defined by ∆ pX := ( id − Φ X ) p ◦ · · · ◦ ( id − Φ X k ) p k , p i ∈ { , } , and Φ X i : B ( H ) → B ( H ) is the completely positive linear map defined by Φ X i ( Y ) := P n i j =1 X i,j Y X ∗ i,j for Y ∈ B ( H ). We use the convention that ( id − Φ f i ,X i ) = id . For information on completely bounded(resp. positive) maps we refer to [14].Let H n i be an n i -dimensional complex Hilbert space. We consider the full Fock space of H n i definedby F ( H n i ) := C ⊕ M p ≥ H ⊗ pn i , where H ⊗ pn i is the (Hilbert) tensor product of p copies of H n i . Let F + n i be the unital free semigroupon n i generators g i , . . . , g in i and the identity g i . Set e iα := e ij ⊗ · · · ⊗ e ij p if α = g ij · · · g ij p ∈ F + n i and e ig i := 1 ∈ C . Note that { e iα : α ∈ F + n i } is an orthonormal basis of F ( H n i ). The length of α ∈ F + n i isdefined by | α | := 0 if α = g i and | α | := p if α = g ij · · · g ij p , where j , . . . , j p ∈ { , . . . , n i } .For each n i ∈ N , i ∈ { , . . . , k } , and j ∈ { , . . . , n i } , let S i,j : F ( H n i ) → F ( H n i ) be the left creationoperator defined by S i,j e iα := e ig j α , α ∈ F + n i , and define the operator S i,j acting on the tensor Hilbertspace F ( H n ) ⊗ · · · ⊗ F ( H n k ) by setting S i,j := I ⊗ · · · ⊗ I | {z } i − ⊗ S i,j ⊗ I ⊗ · · · ⊗ I | {z } k − i times . The k -tuple S := ( S , . . . , S k ), where S i := ( S i, , . . . , S i,n i ), is an element in the regular polyball B n ( ⊗ ki =1 F ( H n i )) and it is called universal model associated with the abstract polyball B n := { B n ( H ) : H is a Hilbert space } . The noncommutative Hardy algebra F ∞ ( B n ) is the sequential SOT-(resp. WOT-, w ∗ -) closure of all polynomials in S i,j and the identity.Let T = ( T , . . . , T k ) ∈ B n ( H ) with T i := ( T i, , . . . , T i,n i ). We use the notation T i,α i := T i,j · · · T i,j p if α i ∈ F + n i and α i = g ij · · · g ij p , and T i,g i := I . The noncommutative Berezin kernel associated with anyelement T in the noncommutative polyball B n ( H ) is the operator K T : H → F ( H n ) ⊗ · · · ⊗ F ( H n k ) ⊗ ∆ T ( I )( H )defined by K T h := X β i ∈ F + ni ,i =1 ,...,k e β ⊗ · · · ⊗ e kβ k ⊗ ∆ T ( I ) / T ∗ ,β · · · T ∗ k,β k h, h ∈ H , where the defect operator is defined by ∆ T ( I ) := ( id − Φ T ) · · · ( id − Φ T k )( I ) . The noncommutativeBerezin kernel K T is a contraction and K T T ∗ i,j = ( S ∗ i,j ⊗ I ) K T for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } .More about the theory of noncommutative Berezin kernels on polybals and polydomains can be found in[17], [20], [21], [22], and [23]. ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 3
In a recent paper [24], we introduced the curvature of any element T ∈ B n ( H ) with trace class defect,i.e. trace [ ∆ T ( I )] < ∞ , by settingcurv ( T ) := lim m →∞ (cid:18) m + kk (cid:19) X q ≥ ,...,qk ≥ q ··· + qk ≤ m trace h K ∗ T ( P (1) q ⊗ · · · ⊗ P ( k ) q k ⊗ I H ) K T i trace h P (1) q ⊗ · · · ⊗ P ( k ) q k i , where K T is the Berezin kernel of T and P ( i ) q i is the orthogonal projection of the full Fock space F ( H n i )onto the span of all vectors e iα i with α ∈ F + n i and | α i | = q i . The curvature is a unitary invariant thatmeasures how far T is from being “free”, i.e. a multiple of the universal model S . We proved the existenceof the curvature and established many asymptotic formulas and its basic properties. These results wereused to develop a theory of curvature (resp. multiplicity) invariant for tensor products of full Fock spaces F ( H n ) ⊗ · · ·⊗ F ( H n k ) and also for tensor products of symmetric Fock spaces F s ( H n ) ⊗ · · · ⊗ F s ( H n k ).Throughout the present paper, unless otherwise specified, we assume that n := ( n , . . . , n k ) ∈ N k with n i ≥
2. Let T = ( T , . . . , T k ) ∈ B ( H ) n × c · · · × c B ( H ) n k be a k -tuple such that its defect operator ∆ T ( I )has finite rank. For each q = ( q , . . . , q k ) ∈ Z k + , where Z + := { } ∪ N , we define the linear manifold M q ( T ) := span (cid:8) T ,α · · · T k,α k h : α i ∈ F + n i , | α i | ≤ q i , h ∈ ∆ T ( I )( H ) (cid:9) . In Section 1, we introduce the
Euler characteristic of T by setting χ ( T ) := lim m →∞ dim M q ( m ) ( T ) Q ki =1 (1 + n i + · · · + n q ( m ) i i ) , where q ( m ) = ( q ( m )1 , . . . , q ( m ) k ) is a cofinal sequence in Z k + . We show the Euler characteristic exists andprovide several asymptotic formulas including the following: χ ( T ) = lim q =( q ,...,q k ) ∈ Z k + rank (cid:2) K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) K T (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) = lim q =( q ,...,q k ) ∈ Z k + rank h ( id − Φ q +1 T ) ◦ · · · ◦ ( id − Φ q k +1 T k )( I ) iQ ki =1 (1 + n i + · · · + n q i i ) , where K T is the Berezin kernel and Φ T , . . . , Φ T k are the completely positive maps associated with T . Itturns out that 0 ≤ curv( T ) ≤ χ ( T ) ≤ rank [ ∆ T ( I )] . We remark that the condition n i ≥ i ∈ { , . . . , k } is needed to prove the existence of the Eulercharacteristic (see the proof of Theorem 1.1). However, there are indications that the result remains trueif one ore more of the n i ’s assume the value one. For now, this remains an open problem. We should addthe fact that the curvature invariant exists if n i ≥ T i,j commutes with T s,t for i = s .We say that M is an invariant subspace of the tensor product F ( H n ) ⊗ · · · ⊗ F ( H n k ) ⊗ H orthat M is invariant under S ⊗ I H if it is invariant under each operator S i,j ⊗ I H . When ( S ⊗ I H ) | M has finite rank defect, we say that M has finite rank. Given two invariant subspaces M and N under S ⊗ I H , we say that they are unitarily equivalent if there is a unitary operator U : M → N such that U ( S i,j ⊗ I H ) | M = ( S i,j ⊗ I H ) | N U .In Section 2, we present a more direct and more transparent proof for the characterization of theinvariant subspaces of ⊗ ki =1 F ( H n i ) with positive defect operator, i.e ∆ M := ∆ S ⊗ I H ( P M ) ≥
0. Thisresult complements the corresponding one from [22] and brings new insight concerning the structure ofthe invariant subspaces of the tensor product ⊗ ki =1 F ( H n i ). The invariant subspaces with positive defectoperators are the so-called Beurling type invariant subspaces .We show that the Euler characteristic completely classifies the finite rank Beurling type invariantsubspaces of S ⊗ I H which do not contain reducing subspaces. In particular, the Euler characteristicclassifies the finite rank Beurling type invariant subspaces of F ( H n ) ⊗ · · · ⊗ F ( H n k ) (see Theorem 2.3). GELU POPESCU
Let M be an invariant subspace of the tensor product F ( H n ) ⊗ · · · ⊗ F ( H n k ) ⊗ E , where E is afinite dimensional Hilbert space. We introduce the Euler characteristic of the orthocomplement M ⊥ bysetting χ ( M ⊥ ) := lim q →∞ · · · lim q k →∞ rank (cid:2) P M ⊥ ( P ≤ ( q ,...,q k ) ⊗ I E ) (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) , where P ≤ ( q ,...,q k ) is the orthogonal projection of ⊗ ki =1 F ( H n i ) onto the span of all vectors of the form e α ⊗ · · · ⊗ e kα k , where α i ∈ F + n i , | α i | ≤ q i . In Section 2, we show that the Euler characteristic of M ⊥ exists and satisfies the equation χ ( M ⊥ ) = χ ( M ) , where M is the compression of S ⊗ I E to the orthocomplement of M . This is used to show that forany t ∈ [0 ,
1] there is an invariant subspace M ( t ) of F ( H n ) ⊗ · · · ⊗ F ( H n k ) such that χ ( M ( t ) ⊥ ) = t and, consequently, the range of the Euler characteristic coincides with the interval [0 , ∞ ). If k = 1, thecorresponding result was proved in [18] and [12]. Moreover, if k ≥
2, we prove that for each t ∈ (0 , { T ( ω ) ( t ) } ω ∈ Ω of non-isomorphic pure elements of rank one defect inthe regular polyball such that χ ( T ( ω ) ( t )) = t, for all ω ∈ Ω . In Section 3, we provide a characterization of the graded invariant subspaces of F ( H n ) ⊗· · ·⊗ F ( H n k )with positive defect operator ∆ M . We also prove that if M is any graded invariant subspace of the tensorproduct F ( H n ) ⊗ · · · ⊗ F ( H n k ), thenlim q →∞ · · · lim q k →∞ rank (cid:2) P M ⊥ P ≤ ( q ,...,q k ) (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) = lim q →∞ · · · lim q k →∞ trace (cid:2) P M ⊥ P ≤ ( q ,...,q k ) (cid:3) trace (cid:2) P ≤ ( q ,...,q k ) (cid:3) . This is equivalent to χ ( P M ⊥ S | M ⊥ ) = curv ( P M ⊥ S | M ⊥ ) , which is an analogue of Arveson’s version [2] of the Gauss-Bonnet-Chern theorem from Riemanniangeometry in our setting. We recall [22] that any pure element with rank-one defect in the polyball B n ( H )has the form P M ⊥ S | M ⊥ , where M is an invariant subspace of the tensor product F ( H n ) ⊗· · ·⊗ F ( H n k ).It remains an open problem whether, as in the commutative setting ([6], [7]), the result above holds truefor any invariant subspace M generated by arbitrary polynomials. We mention that a version of theGauss-Bonnet-Chern theorem for graded pure elements of finite rank defect in the noncommutativepolyball is also obtained. Finally, we remark that the results of this paper can be re-formulated in termsof Hilbert modules [5] over the complex semigroup algebra C [ F + n × · · · × F + n k ] generated by direct productof free semigroups. In this setting, the Hilbert module associated with the universal model of the abstractpolyball B n plays the role of rank-one free module in the algebraic theory [11].1. Euler characteristic on noncommutative polyballs
In this section we prove the existence of the Euler characteristic of any element T in the noncommuta-tive polyball and provide several asymptotic formulas in terms of the noncommutative Berezin kernel andthe completely positive maps associated with T ∈ B n ( H ). Basic properties of the Euler characteristicare also proved.Let T = ( T , . . . , T k ) ∈ B ( H ) n × c · · · × c B ( H ) n k with T i := ( T i, , . . . , T i,n i ), and let D ⊂ H be a finitedimensional subspace. For each q = ( q , . . . , q k ) ∈ Z k + , where Z + = { , , . . . } , we define M q ( T , D ) := span (cid:8) T ,α · · · T k,α k h : α i ∈ F + n i , | α i | ≤ q i , h ∈ D (cid:9) . We also use the notation M ( q ,...,q k ) when T and D are understood and we want to emphasis the coordi-nates q , . . . q k . Given two k -tuples q = ( q , . . . , q k ) and p = ( p , . . . , p k ) in Z k + , we consider the partialorder q ≤ p defined by q i ≤ p i for any i ∈ { , . . . , k } . We consider Z k + as a directed set with respect tothis partial order. ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 5
Theorem 1.1.
Let
D ⊂ H be a finite dimensional subspace and let T ∈ B ( H ) n × c · · · × c B ( H ) n k , where n i ≥ for any i ∈ { , . . . , k } . Then the following iterated limits exist and are equal lim q σ (1) →∞ · · · lim q σ ( k ) →∞ dim M q ( T , D ) n q · · · n q k k , where σ is any permutation of { , . . . , k } and q = ( q , . . . , q k ) ∈ Z k + . Moreover, these limits coincide with lim q =( q ,...,q k ) ∈ Z k + dim M q ( T , D ) n q · · · n q k k . Proof. If q = ( q , . . . , q k ) ∈ Z k + , we use the notation M ( q ,...,q k ) := M q ( T , D ). For i ∈ { , . . . , k } , notethat M ( q ,...,q k ) = M ( q ,...q i − , ,q i +1 ...,q k ) + T i, M ( q ,...q i − ,q i − ,q i +1 ...,q k ) + · · · + T i,n i M ( q ,...q i − ,q i − ,q i +1 ...,q k ) . Consequently, we deduce that(1.1) dim M ( q ,...,q k ) ≤ n i dim M ( q ,...q i − ,q i − ,q i +1 ...,q k ) + dim M ( q ,...q i − , ,q i +1 ...,q k ) for any ( q , . . . , q k ) ∈ Z k + . Iterating this inequality, we obtain(1.2) dim M ( q ,...,q k ) ≤ (1 + n i + · · · + n q i i ) dim M ( q ,...q i − , ,q i +1 ...,q k ) , which implies(1.3) dim M ( q ,...,q k ) Q ki =1 (1 + n i + · · · + n q i i ) ≤ dim M (0 ,..., = dim D for any ( q , . . . , q k ) ∈ Z k + . On the other hand, relations (1.1), (1.2), and (1.3) imply(1.4) 0 ≤ dim M ( q ,...,q k ) n q i i ≤ dim M ( q ,...q i − ,q i − ,q i +1 ...,q k ) n q i − i + 1 n q i i k Y s =1 s = i (1 + n s + · · · + n q s s ) dim D . For x ∈ R we set x + := max { x, } and x − := max {− x, } . Denote x q i := dim M ( q ,...,qk ) n qii and note that0 ≤ x N = x + N X p =1 ( x p − x p − ) + − N X p =1 ( x p − x p − ) − . Hence, using relation (1.4), we deduce that N X p =1 ( x p − x p − ) − ≤ x + N X p =1 ( x p − x p − ) + ≤ x + N X p =1 n pi ! k Y s =1 s = i (1 + n s + · · · + n q s s ) dim D . Consequently, we obtain x + N X p =1 | x p − x p − | ≤ x + 2 N X p =1 ( x p − x p − ) + for any N ∈ N . Since n i ≥
2, the inequalities above show that the sequence x N = x + P Np =1 ( x p − x p − )is convergent as N → ∞ . Therefore,(1.5) lim q i →∞ dim M ( q ,...,q k ) n q i i exists for any q , . . . q i − , q i +1 . . . , q k ∈ Z + . The next step in our proof is to show that the iterated limitlim q k →∞ · · · lim q →∞ dim M ( q ,...,q k ) n q · · · n q k k exists. GELU POPESCU
We use an inductive argument. Due to relation (1.5), the limit lim q →∞ dim M ( q ,...,qk ) n q exists for any q , . . . , q k in Z + . Assume that 1 ≤ p ≤ k − y ( q p +1 , . . . , q k ) := lim q p →∞ · · · lim q →∞ dim M ( q ,...,q k ) n q · · · n q p p exists for any q p +1 , . . . , q k in Z + . Due to relation (1.4), we have0 ≤ dim M ( q ,...,q k ) n q · · · n q p p ≤ n p +1 dim M ( q ,...q p ,q p +1 − ,q p +2 ...,q k ) n q · · · n q p p + 1 n q · · · n q p p k Y s =1 s = p +1 (1 + n s + · · · + n q s s ) dim D . Consequently, taking the limits as q → ∞ , . . . , q p → ∞ , we deduce that y ( q p +1 , . . . , q k ) n q p +1 p +1 ≤ y ( q p +1 − , q p +2 , . . . , q k ) n q p +1 − p +1 + 1 n q p +1 p +1 p Y s =1 n s n s − k Y s = p +2 (1 + n s + · · · + n q s s ) dim D . Setting z q p +1 := y ( q p +1 ,...,q k ) n qp +1 p +1 , we have0 ≤ z q p +1 ≤ z q p +1 − + 1 n q p +1 p +1 p Y s =1 n s n s − k Y s = p +2 (1 + n s + · · · + n q s s ) dim D , q p +1 ∈ N . Similarly to the proof that x q i is convergent (see relation (1.5)), one can show that the sequence { z q p +1 } is convergent as q p +1 → ∞ for any q p +2 , . . . , q k in Z + . Therefore, the iterated limit y ( q p +2 , . . . , q k ) := lim q p +1 →∞ · · · lim q →∞ dim M ( q ,...,q k ) n q · · · n q p +1 p +1 exists , which proves our assertion. Since the entries of X s := ( X s, , . . . , X s,n s ) are commuting with the entriesof X t := ( X t, , . . . , X t,n t ) for any s, t ∈ { , . . . , k } , s = t , similar arguments as above show that, for anypermutation σ of { , . . . , k } , the iterated limit L σ := lim q σ (1) →∞ · · · lim q σ ( k ) →∞ a ( q ,...,q k ) exists, where a ( q ,...,q k ) := dim M ( q ,...,qk ) n q ··· n qkk .The next step is to show that all these iterated limits are equal. We proceed by contradiction. Withoutloss of generality we can assume that L id < L σ for some permutation σ . Let ǫ > ǫ < L σ − L id . Since n i ≥ L id = lim q →∞ · · · lim q k →∞ a ( q ,...,q k ) , we can choose natural numbers N ≤ N ≤ . . . ≤ N k such that P ∞ j = N i +1 1 n ji < ǫkM for any i ∈ { , . . . , k } , where M := (dim D ) Q ki =1 n i n i − ,and such that | a ( N ,...,N k ) − L id | < ǫ. Taking into account that L σ = lim q σ (1) →∞ · · · lim q σ ( k ) →∞ a ( q ,...,q k ) , we can choose natural numbers C , . . . , C k such that C i > max { N , . . . , N k } and | a ( C ,...,C k ) − L σ | < ǫ. Consequently, we obtain(1.6) a ( C ,...,C k ) − a ( N ,...,N k ) > ǫ. For each ( q , . . . , q k ) ∈ Z k + , set d ( q ,...,q k ) := 1 n q · · · n q k k k Y i =1 (1 + n i + · · · n q i i ) dim D ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 7 and note that d ( q ,...,q k ) ≤ (dim D ) Q ki =1 n i n i − . Using relation (1.1), we deduce that a ( C ,C ,...,C k ) = a ( N ,N ,...,N k ) + [( a ( N +1 ,N ,...,N k ) − a ( N ,N ,...,N k ) ) + · · · + ( a ( C ,N ,...,N k ) − a ( C − ,N ,...,N k ) )]+ [( a ( C ,N +1 ,N ,...,N k ) − a ( C ,N ,N ,...,N k ) ) + · · · + ( a ( C ,C ,N ,...,N k ) − a ( C ,C − ,N ...,N k ) )]...+ [( a ( C ,C ,...,C k − ,N k +1) − a ( C ,C ,...,C k − ,N k ) ) + · · · + ( a ( C ,C ,...,C k ) − a ( C ,C ,...,C k − ,C k − )] ≤ a ( N ,N ,...,N k ) + C X j = N +1 n j dim M (0 ,N ,...,N k ) n N · · · n N k k + C X j = N +1 n j dim M ( C , ,N ,...,N k ) n C n N · · · n N k k + · · · + C k X j = N k +1 n jk dim M ( C , ,N ,...,N k ) n C n N · · · n N k − k − ≤ a ( N ,N ,...,N k ) + d (0 ,N ,...,N k ) C X j = N +1 n j + d ( C , ,N ,...,N k ) C X j = N +1 n j + · · · + d ( C ,...,C k − , C k X j = N k +1 n j ≤ a ( N ,N ,...,N k ) + ǫkM k (dim D ) k Y i =1 n i n i − a ( N ,N ,...,N k ) + ǫ. Consequently, we have 0 ≤ a ( C ,C ,...,C k ) − a ( N ,N ,...,N k ) ≤ ǫ, which contradicts relation (1.6). Therefore, we must have L id = L σ . This completes the proof that theiterated limits lim q σ (1) →∞ · · · lim q σ ( k ) →∞ a ( q ,...,q k ) exist and are equal for any permutation σ of the set { , . . . , k } .Now, we show that the net { a ( q ,...,q k ) } ( q ,...,q k ) ∈ Z k + is convergent andlim ( q ,...,q k ) ∈ Z k + a ( q ,...,q k ) = L id . Assume, by contradiction, that the net { a ( q ,...,q k ) } ( q ,...,q k ) ∈ Z k + is not convergent to L id . Then we canfind ǫ > q ∈ Z k + there exists p ∈ Z k + with p ≥ q such that a p ∈ ( −∞ , L id − ǫ ] ∪ [ L id + 3 ǫ , ∞ ) . In particular, for each q ( n ) = ( n, . . . , n ) ∈ Z k + there exists p ( n ) ∈ Z k + with p ( n ) > q ( n ) and such that therelation above holds. This implies that at least one of the setsΓ − := { q ∈ Z k + : a q ≤ L − ǫ } and Γ + := { q ∈ Z k + : a q ≥ L + 3 ǫ } is cofinal in Z k + . Assume that Γ − is cofinal in Z k + . Then, using that n i ≥ i ∈ { , . . . , k } , wecan find natural numbers D , . . . , D k such that P ∞ j = D i +1 1 n ji < ǫ kM for i ∈ { , . . . , k } , where M :=(dim D ) Q ki =1 n i n i − , and such that a ( D ,...,D k ) ≤ L id − ǫ . Since L id = lim q →∞ · · · lim q k →∞ a ( q ,...,q k ) , we can choose natural numbers R , . . . , R k such that R i > C i for i ∈ { , . . . , k } and such that a ( R ,...,R k ) ∈ ( L id − ǫ , L + ǫ ) . Consequently, we deduce that a ( R ,...,R k ) − a ( D ,...,D k ) ≥ ǫ . On the other hand, as in the first part ofthe proof, one can show that a ( R ,...,R k ) ≤ a ( D ,...,D k ) + ǫ , GELU POPESCU which is a contradiction.Now, assume that Γ + is a cofinal set in Z k + . Since n i ≥ L id = lim q →∞ · · · lim q k →∞ a ( q ,...,q k ) ,we can choose natural numbers N ≤ N ≤ . . . ≤ N k such that P ∞ j = N i +1 1 n ji < ǫkM for any i ∈ { , . . . , k } ,where M := (dim D ) Q ki =1 n i n i − , and such that | a ( N ,...,N k ) − L id | < ǫ. Taking into account that Γ + is acofinal set in Z k + , we can find natural numbers N ′ , . . . , N ′ k such that N ′ i ≥ N i for any i ∈ { , . . . , k } andsuch that a ( N ′ ,...,N ′ k ) ≥ L id + 3 ǫ . Consequently, we have a ( N ′ ,...,N ′ k ) − a ( N ,...,N k ) ≥ ǫ . Once again, as in the first part of the proof,since( N ′ , . . . , N ′ k ) ≥ ( N , . . . , N k ) one can show that a ( N ′ ,...,N ′ k ) − a ( N ,...,N k ) ≤ ǫ , which is a contradiction.This completes the proof. (cid:3) Let T = ( T , . . . , T k ) ∈ B ( H ) n × c · · · × c B ( H ) n k be a k -tuple such that its defect operator defined by ∆ T ( I ) := ( id − Φ T ) · · · ( id − Φ T k )( I ) has finite rank. In this case, we say that T has finite rank defector that T has finite rank. For each q = ( q , . . . , q k ) ∈ Z k + , we define M q ( T ) := span (cid:8) T ,α · · · T k,α k h : α i ∈ F + n i , | α i | ≤ q i , h ∈ ∆ T ( I )( H ) (cid:9) . Let { q ( m ) } ∞ m =1 with q ( m ) = ( q ( m )1 , . . . , q ( m ) k ) be an increasing cofinal sequence in Z k + . Then M q (1) ( T ) ⊆ M q (2) ( T ) ⊆ · · · ⊆ M ( T ) and ∞ [ m =1 M q ( m ) ( T ) = M ( T ) , where M ( T ) := span n T ,α · · · T k,α k h : α i ∈ F + n i , h ∈ ∆ T ( I ) / ( H ) o . We introduce the
Euler characteristic of T by setting χ ( T ) := lim m →∞ dim M q ( m ) ( T ) Q ki =1 (1 + n i + · · · + n q ( m ) i i ) . We remark that the Euler characteristic does not depend on any topology associated with the Hilbertspace H but depends solely on the linear algebra of the linear submanifold M ( T ) of H .For each q i ∈ { , , . . . } and i ∈ { , . . . , k } , let P ( i ) q i be the orthogonal projection of the full Fock space F ( H n i ) onto the span of all vectors e iα i with α ∈ F + n i and | α i | = q i . We recall that P ≤ ( q ,...,q k ) is theorthogonal projection of ⊗ ki =1 F ( H n i ) onto the span of all vectors of the form e α ⊗ · · · ⊗ e kα k , where α i ∈ F + n i , | α i | ≤ q i . In what follows, we prove the existence of the Euler characteristic associated witheach element in the regular polyball and establish several asymptotic formulas. Theorem 1.2.
Let T = ( T , . . . , T k ) be in the regular polyball B n ( H ) and have finite rank defect. If n := ( n , . . . , n k ) with n i ≥ , then the Euler characteristic of T exists and satisfies the asymptoticformulas χ ( T ) = lim q =( q ,...,q k ) ∈ Z k + dim M q ( T ) n q · · · n q k k k Y i =1 (cid:18) − n i (cid:19) = lim q =( q ,...,q k ) ∈ Z k + rank (cid:2) K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) K T (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) = lim q =( q ,...,q k ) ∈ Z k + rank h ( id − Φ q +1 T ) ◦ · · · ◦ ( id − Φ q k +1 T k )( I ) iQ ki =1 (1 + n i + · · · + n q i i ) , where K T is the Berezin kernel and Φ T , . . . , Φ T k are the completely positive maps associated with T . ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 9
Proof.
Applying Theorem 1.1 when D = ∆ T ( I ) / ( H ), we deduce that lim m →∞ dim M q ( m ) ( T ) n q ( m )11 ··· n q ( m ) kk exists anddoes not depend on the choice of the cofinal sequence { q ( m ) = ( q ( m )1 , . . . , q ( m ) k ) } ∞ m =1 in Z k + . Moreover,the limit coincides with lim q =( q ,...,q k ) ∈ Z k + dim M q ( T ) n q · · · n q k k = lim q →∞ · · · lim q k →∞ dim M q ( T ) n q · · · n q k k . Consequently, the Euler characteristic of T exists and the first equality in the theorem holds. Thenoncommutative Berezin kernel associated with T ∈ B n ( H ) is the operator K T : H → F ( H n ) ⊗ · · · ⊗ F ( H n k ) ⊗ ∆ T ( I )( H )defined by K T h := X β i ∈ F + ni ,i =1 ,...,k e β ⊗ · · · ⊗ e kβ k ⊗ ∆ T ( I ) / T ∗ ,β · · · T ∗ k,β k h, h ∈ H . It is easy to see that K ∗ T ( e β ⊗ · · · ⊗ e kβ k ⊗ h ) = T ,β · · · T k,β k ∆ T ( I ) / h for any β i ∈ F + n i and i ∈ { , . . . , k } . This shows that the range of K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) is equal to M ( q ,...,q k ) ( T ) := M q ( T ). Hence, the range of K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) K T is equal to M q ( T ). Sincerank (cid:2) P ≤ ( q ,...,q k ) (cid:3) = k Y i =1 (1 + n i + · · · + n q i i ) , the results above imply χ ( T ) = lim q →∞ · · · lim q k →∞ rank (cid:2) K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) K T (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) . The next step in our proof is to show that(1.7) K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) K T = ( id − Φ q +1 T ) ◦ · · · ◦ ( id − Φ q k +1 T k )( I )for any ( q , . . . , q k ) ∈ Z k + . Let S := ( S , . . . , S k ) ∈ B n ( ⊗ ki =1 F ( H n i )) with S i := ( S i, , . . . , S i,n i ) be theuniversal model associated with the abstract polyball B n . It is easy to see that S := ( S , . . . , S k ) is apure k -tuple, i.e. Φ p S i ( I ) → p → ∞ , and( id − Φ S ) ◦ · · · ◦ ( id − Φ S k )( I ) = P C , where P C is the orthogonal projection from ⊗ ki =1 F ( H n i ) onto C ⊂ ⊗ ki =1 F ( H n i ), where C C ⊗ · · · ⊗ C
1. Taking into account that K T T ∗ i,j = ( S ∗ i,j ⊗ I ) K T for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } , we deduce that K ∗ T ( P (1) q ⊗ · · · ⊗ P ( k ) q k ⊗ I H ) K T = K ∗ T h (Φ q S − Φ q +1 S ) ◦ · · · ◦ (Φ q k S k − Φ q k +1 S k )( I ) ⊗ I H i K T = K ∗ T (cid:2) Φ q S ◦ · · · ◦ Φ q k S k ◦ ( id − Φ S k ) ◦ · · · ◦ ( id − Φ S )( I ) ⊗ I H (cid:3) K T = Φ q T ◦ · · · ◦ Φ q k T k ◦ ( id − Φ T ) ◦ · · · ◦ ( id − Φ T k )( K ∗ T K T ) . On the other hand, since(1.8) K ∗ T K T = lim q k →∞ . . . lim q →∞ ( id − Φ q k T k ) ◦ · · · ◦ ( id − Φ q T )( I ) , where the limits are in the weak operator topology, we can prove that(1.9) ( id − Φ T ) ◦ · · · ◦ ( id − Φ T k )( K ∗ T K T ) = ∆ T ( I ) . Indeed, note that { ( id − Φ q k T k ) ◦ · · · ◦ ( id − Φ q T )( I ) } q =( q ,...,q k ) ∈ Z k + is an increasing sequence of positiveoperators and( id − Φ q k T k ) ◦ · · · ◦ ( id − Φ q T )( I ) = q k − X s k =0 Φ s k T k ◦ · · · q − X s =0 Φ s T ◦ ( id − Φ T k ) ◦ · · · ◦ ( id − Φ T )( I ) . Since Φ T , . . . , Φ T k are commuting WOT-continuous completely positive linear maps and lim q i →∞ Φ q i T i ( I )exists in the weak operator topology for each i ∈ { , . . . , k } , we have( id − Φ T )( K ∗ T K T ) = lim q k →∞ . . . lim q →∞ ( id − Φ q k T k ) ◦ · · · ◦ ( id − Φ q T ) ◦ ( id − Φ T )( I )= lim q k →∞ . . . lim q →∞ ( id − Φ q k T k ) ◦ · · · ◦ ( id − Φ q T ) (cid:20) lim q →∞ ( id − Φ q T ) ◦ ( id − Φ T )( I ) (cid:21) = lim q k →∞ . . . lim q →∞ ( id − Φ q k T k ) ◦ · · · ◦ ( id − Φ q T ) ◦ ( id − Φ T )( I ) . Applying now id − Φ T , a similar reasoning leads to( id − Φ T ) ◦ ( id − Φ T )( K ∗ T K T ) = lim q k →∞ . . . lim q →∞ ( id − Φ q k T k ) ◦ · · · ◦ ( id − Φ q T ) ◦ ( id − Φ T ) ◦ ( id − Φ T )( I ) . Continuing this process, we obtain relation (1.9). Combining (1.9) with the relation preceding (1.8), wehave K ∗ T ( P (1) q ⊗ · · · ⊗ P ( k ) q k ⊗ I ) K T = Φ q T ◦ · · · ◦ Φ q k T k ( ∆ T ( I ))for any q , . . . , q k ∈ Z + , which implies K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) K T = q k X s k =0 Φ s k T k ◦ · · · q X s =0 Φ s T ◦ ( id − Φ T k ) ◦ · · · ◦ ( id − Φ T )( I )= ( id − Φ q +1 T ) ◦ · · · ◦ ( id − Φ q k +1 T k )( I )and proves relation (1.7). Now, one can easily complete the proof. (cid:3) We remark that, in Theorem 1.2, the limit over q = ( q , . . . , q k ) ∈ Z k + can be replaced by the limitover any increasing cofinal sequence in Z k + , or by any iterated limit lim q σ (1) →∞ · · · lim q σ ( k ) →∞ , where σ isa permutation of { , . . . , k } . Corollary 1.3. If T ∈ B n ( H ) has finite rank defect, then ≤ curv( T ) ≤ χ ( T ) ≤ rank [ ∆ T ( I )] . Proof.
According to [24], we havecurv( T ) = lim q →∞ · · · lim q k →∞ trace (cid:2) K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I H ) K T (cid:3) trace (cid:2) P ≤ ( q ,...,q k ) (cid:3) . Since trace h P (1) q ⊗ · · · ⊗ P ( k ) q k i = rank h P (1) q ⊗ · · · ⊗ P ( k ) q k i and K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I H ) K T is a positivecontraction, Theorem 1.2 implies curv( T ) ≤ χ ( T ). The inequality χ ( T ) ≤ rank [ ∆ T ( I )] is due to theinequality (1.3), in the particular case when D = ∆ T ( I ) / ( H ). (cid:3) Corollary 1.4. If T ∈ B n ( H ) and T ′ ∈ B n ( H ′ ) have finite rank defects, then T ⊕ T ′ ∈ B n ( H ⊕ H ′ ) hasfinite rank defect and χ ( T ⊕ T ′ ) = χ ( T ) + χ ( T ′ ) . If, in addition, dim H ′ < ∞ , then χ ( T ⊕ T ′ ) = χ ( T ) . Given a function κ : N → N and n ( i ) := ( n ( i )1 , . . . , n ( i ) κ ( i ) ) ∈ N κ ( i ) for i ∈ { , . . . , p } , we consider thepolyball B n ( i ) ( H i ), where H i is a Hilbert space. Let X ( i ) ∈ B n ( i ) ( H i ) with X ( i ) := ( X ( i )1 , . . . , X ( i ) κ ( i ) ) and X ( i ) r := ( X ( i ) r, , . . . , X ( i ) r,n ( i ) r ) ∈ B ( H i ) n ( i ) r for r ∈ { , . . . , κ ( i ) } . If X := ( X (1) , . . . , X ( p ) ) ∈ B n (1) ( H ) × · · · × B n ( p ) ( H p ), we define the ampliation e X by setting e X := ( e X (1) , . . . , e X ( p ) ), where e X ( i ) := ( e X ( i )1 , . . . , e X ( i ) κ ( i ) )and e X ( i ) r := ( e X ( i ) r, , . . . , e X ( i ) r,n ( i ) r ) for r ∈ { , . . . , κ ( i ) } , and e X ( i ) r,s := I H ⊗ · · · ⊗ I H i − ⊗ X ( i ) r,s ⊗ I H i +1 ⊗ I H p for all i ∈ { , . . . , p } , r ∈ { , . . . , κ ( i ) } , and s ∈ { , . . . , n ( i ) r } . ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 11
Theorem 1.5.
Let X := ( X (1) , . . . , X ( p ) ) ∈ B n (1) ( H ) × · · · × B n ( p ) ( H p ) be such that each X ( i ) has finiterank defect. Then the ampliation e X is in the regular polyball B ( n (1) ,..., n ( p ) ) ( H ⊗ · · · ⊗ H p ) , has finite rankdefect, and the Euler characteristic satisfies the relation χ ( e X ) = p Y i =1 χ ( X ( i ) ) . Proof.
For each m ( i ) ∈ Z κ ( i )+ with 0 ≤ m ( i ) ≤ (1 , . . . ,
1) and i ∈ { , . . . , p } , we have ∆ ( m (1) ,..., m ( p ) ) e X ( I H ⊗···⊗H p ) = ∆ m (1) X (1) ( I H ) ⊗ · · · ⊗ ∆ m ( p ) X ( p ) ( I H p ) ≥ . Consequently, e X is in the regular polyball B ( n (1) ,..., n ( p ) ) ( H ⊗ · · · ⊗ H p ) andrank (cid:2) ∆ e X ( I H ⊗···⊗H p ) (cid:3) = rank [ ∆ X (1) ( I H )] · · · rank (cid:2) ∆ X ( p ) ( I H p ) (cid:3) < ∞ . Let q ( i ) := ( q ( i )1 , . . . , q ( i ) κ ( i ) ) ∈ Z κ ( i )+ for i ∈ { , . . . , p } . According to Theorem 1.2, the Euler characteristic χ ( e X ) is equallim rank (cid:20) ( id − Φ q (1)1 +1 X (1)1 ) ◦ · · · ◦ ( id − Φ q (1) κ (1) +1 X (1) κ (1) )( I H ) ⊗ · · · ⊗ ( id − Φ q ( p )1 +1 X ( p )1 ) ◦ · · · ◦ ( id − Φ q ( p ) κ ( p ) +1 X ( p ) κ ( p ) )( I H p ) (cid:21)Q κ (1) r =1 (1 + n (1) r + · · · + ( n (1) r ) q (1) r ) · · · Q κ ( p ) r =1 (1 + n ( p ) r + · · · + ( n ( p ) r ) q ( p ) r ) , where the limit is taken over q (1) ∈ Z κ (1)+ , . . . , q ( p ) ∈ Z κ ( p )+ , which is equal to to the product p Y i =1 lim q ( i ) ∈ Z κ ( i )+ rank (cid:20) ( id − Φ q ( i )1 +1 X ( i )1 ) ◦ · · · ◦ ( id − Φ q ( i ) κ ( i ) +1 X ( i ) κ ( i ) )( I H i ) (cid:21)Q κ ( i ) r =1 (1 + n ( i ) r + · · · + ( n ( i ) r ) q ( i ) r ) . Due to Theorem 1.2, the latter product is equal to Q pi =1 χ ( X ( i ) ). The proof is complete. (cid:3) Invariant subspaces, Euler characteristic, and classification
In this section, we give a characterization of the invariant subspaces of the tensor product ⊗ ki =1 F ( H n i )with positive defect operators. We show that the Euler characteristic completely classifies the finite rankBeurling type invariant subspaces of ⊗ ki =1 F ( H n i ) and prove some of its basic properties, including thefact that its range is the interval [0 , ∞ ).Let S := ( S , . . . , S k ), where S i := ( S i, , . . . , S i,n i ), be the universal model of the abstract polyball B n , and let H be a Hilbert space. We say that M is an invariant subspace of ⊗ ki =1 F ( H n i ) ⊗ H orthat M is invariant under S ⊗ I H if it is invariant under each operator S i,j ⊗ I H for i ∈ { , . . . , k } and j ∈ { , . . . , n j } . Definition 2.1.
Given two invariant subspaces M and N under S ⊗ I H , we say that they are unitarilyequivalent if there is a unitary operator U : M → N such that U ( S i,j ⊗ I H ) | M = ( S i,j ⊗ I H ) | N U for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } . We define the right creation operator R i,j : F ( H n i ) → F ( H n i ) by setting R i,j e iα := e iαg j , α ∈ F + n i , and the operator R i,j acting on the tensor Hilbert space F ( H n ) ⊗ · · · ⊗ F ( H n k ). We also denote R := ( R , . . . , R k ), where R i := ( R i, , . . . , R i,n i ). Let ϕ = P β i ∈ F + ni c β ,...,β k e β ⊗ · · · ⊗ e kβ k be in ⊗ ki =1 F ( H n i ) and consider the formal power series e ϕ ( R i,j ) := P β i ∈ F + ni c β ,...,β k R , e β · · · R k, e β k , where e β i = g ij p · · · g ij denotes the reverse of β i = g ij · · · g ij p ∈ F + n i . Note that e ϕ ( R i,j )( e γ ⊗ · · · ⊗ e kγ k ) is in ⊗ ki =1 F ( H n i ). We say that ϕ is a right multiplier of ⊗ ki =1 F ( H n i ) ifsup p ∈P , k p k≤ k e ϕ ( R i,j ) p k < ∞ , where P is the set of all polynomials P a α ,...,α k e α ⊗ · · · ⊗ e kα k in ⊗ ki =1 F ( H n i ). In this case, there is aunique bounded operator acting on ⊗ ki =1 F ( H n i ), denoted also by e ϕ ( R i,j ), such that e ϕ ( R i,j ) p = X β i ∈ F + ni c β ,...,β k R , e β · · · R k, e β k p, p ∈ P . The set of all operators e ϕ ( R i,j ) satisfying the conditions above is a Banach algebra denoted by R ∞ ( B n ).We proved in [22] that F ∞ ( B n ) ′ = R ∞ ( B n ) and F ∞ ( B n ) ′′ = F ∞ ( B n ), where ′ stands for the commutant.According to Theorem 5.1 from [22], if a subspace M ⊂ ⊗ ki =1 F ( H n i ) ⊗ H is co-invariant under eachoperator S i,j ⊗ I H , thenspan (cid:8) ( S ,β · · · S k,β k ⊗ I H ) M : β ∈ F + n , . . . , β k ∈ F + n k (cid:9) = ⊗ ki =1 F ( H n i ) ⊗ E , where E := ( P C ⊗ I H )( M ). Consequently, a subspace R ⊆ ⊗ ki =1 F ( H n i ) ⊗ H is reducing under S ⊗ I H if and only if there exists a subspace G ⊆ H such that R = ⊗ ki =1 F ( H n i ) ⊗ G . It is well known [26] that the lattice of the invariant subspaces for the Hardy space H ( D k ) is verycomplicated and contains many invariant subspaces which are not of Beurling type. The same complicatedsituation occurs in the case of the tensor product ⊗ ki =1 F ( H n i ). Following the classical case [4], we saythat M is a Beurling type invariant subspace for S ⊗ I H if there is an inner multi-analytic operatorΨ : ⊗ ki =1 F ( H n i ) ⊗ E → ⊗ ki =1 F ( H n i ) ⊗ H with respect to S , where E is a Hilbert space, such that M = Ψ (cid:2) ⊗ ki =1 F ( H n i ) ⊗ E (cid:3) . In this case, Ψ can be chosen to be isometric. We recall that Ψ is multi-analytic if Ψ( S i,j ⊗ I H ) = ( S i,j ⊗ I E )Ψ for all i, j . More about multi-analytic operators on Fock spacescan be found in [15] and [16]. In [22], we proved that M is a Beurling type invariant subspace for S ⊗ I H if and only if ( id − Φ S ⊗ I H ) ◦ · · · ◦ ( id − Φ S k ⊗ I H )( P M ) ≥ , where P M is the orthogonal projection onto M . We introduce the defect operator of M be setting∆ M := ∆ S ⊗ I H ( P M ).In what follows, we present a more direct and more transparent proof for the characterization of theinvariant subspaces of ⊗ ki =1 F ( H n i ) with positive defect operator, which complements the correspondingresult from [22] (see Corollary 5.3). Theorem 2.2.
A subspace
M ⊆ ⊗ ki =1 F ( H n i ) , n i ≥ , is invariant under the universal model S andhas positive defect operator ∆ M if and only if there is a sequence { ψ s } Ns =1 , where N ∈ N or N = ∞ , ofright multipliers of ⊗ ki =1 F ( H n i ) such that { e ψ s ( R i,j ) } Ns =1 are isometries with orthogonal ranges and P M = N X s =1 e ψ s ( R i,j ) e ψ s ( R i,j ) ∗ , where the convergence is in the strong operator topology. The sequence { ψ s } Ns =1 is uniquely determinedup to a unitary equivalence.In addition, we can choose the sequence { ψ s } Ns =1 such that each ψ s is in the range of ∆ M . We alsohave N = rank (∆ / M ) and ∆ M ξ = N X t =1 h ξ, ψ s i ψ s , ξ ∈ ⊗ ki =1 F ( H n i ) . Proof.
Assume that
M ⊆ ⊗ ki =1 F ( H n i ) is an invariant subspace under S and ∆ M := ∆ S ( P M ) ≥
0. Let A = ( A , . . . , A k ) with A i := ( S i, | M , . . . , S i,n i | M ) and note that A is a pure element in B n ( M ). Thenoncommutative Berezin kernel associated with A , K A : M → (cid:0) ⊗ ki =1 F ( H n i ) (cid:1) ⊗ ∆ A ( I M ) / ( M ) , is an isometry. Note that ∆ M ( M ) ⊆ M and ∆ M | M = ∆ A ( I M ). Consequently, we have ∆ M ( M ⊥ ) ⊆M ⊥ and ∆ A ( I M ) / ( M ) = ∆ / M ( ⊗ ki =1 F ( H n i )). Consider the extended Berezin kernel b K A : ⊗ ki =1 F ( H n i ) → (cid:0) ⊗ ki =1 F ( H n i ) (cid:1) ⊗ ∆ / M ( ⊗ ki =1 F ( H n i )) ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 13 defined by b K A | M := K A and b K A | M ⊥ := 0. Let { χ s } Ns =1 be an orthonormal basis for the defect space∆ / M ( ⊗ ki =1 F ( H n i )) and set ψ s := b K ∗ A (1 ⊗ χ s ) = K ∗ A (1 ⊗ χ s ) = ∆ A ( I M ) / χ s = ∆ / M χ s . If p = P a α ,...,α k e α ⊗ · · · ⊗ e kα k is any polynomial in ⊗ ki =1 F ( H n i ), we have K ∗ A ( p ⊗ χ s ) = p ( S i,j | M )∆ / M χ s = p ( S i,j ) ψ s = pψ s = e ψ s ( R i,j ) p. Hence, k e ψ s ( R i,j ) p k = k K ∗ A ( p ⊗ χ s ) k ≤ k p k . Since the polynomials are dense in ⊗ ki =1 F ( H n i ), we deduce that ψ s is a right multiplier of ⊗ ki =1 F ( H n i ).For each s ∈ { , . . . , N } , we define the linear operator Γ s : ⊗ ki =1 F ( H n i ) → ⊗ ki =1 F ( H n i ) by setting h Γ s f, g i := D b K A f, g ⊗ χ s E , f, g ∈ ⊗ ki =1 F ( H n i ) . Note that N X s =1 k Γ s f k = N X s =1 X α i ∈ F + ni | (cid:10) Γ s f, e α ⊗ · · · ⊗ e kα k (cid:11) | = N X s =1 X α i ∈ F + ni (cid:12)(cid:12)(cid:12)D b K A f, ( e α ⊗ · · · ⊗ e kα k ) ⊗ χ s E(cid:12)(cid:12)(cid:12) = k b K A f k ≤ k K A f k = k f k for any f ∈ ⊗ ki =1 F ( H n i ). Consequently, P Ns =1 Γ ∗ s Γ s is convergent in the strong operator topology and N X s =1 Γ ∗ s Γ s = b K ∗ A b K A = (cid:20) K ∗ A K A
00 0 (cid:21) = (cid:20) I M
00 0 (cid:21) = P M . Note also that h Γ s f, p i = h f, K ∗ A ( p ⊗ χ s ) i = D f, e ψ s ( R i,j ) p E = D e ψ s ( R i,j ) ∗ f, p E . Since the polynomials are dense in ⊗ ki =1 F ( H n i ), we deduce that Γ s = e ψ s ( R i,j ) ∗ and, consequently, P Ns =1 e ψ s ( R i,j ) e ψ s ( R i,j ) ∗ = P M , where the convergence is in the strong operator topology. On the otherhand, since Φ S i is WOT-continuous, so is the defect map ∆ S := ( id − Φ S ) ◦· · ·◦ ( id − Φ S k ). Consequently,∆ M = ∆ S ( P M ) = N X s =1 e ψ s ( R i,j ) ∆ S ( I ) e ψ s ( R i,j ) ∗ = N X s =1 e ψ s ( R i,j ) P C e ψ s ( R i,j ) ∗ . Hence, we deduce that∆ M ξ = N X s =1 e ψ s ( R i,j ) D e ψ s ( R i,j ) ∗ ξ, E = N X t =1 h ξ, ψ s i ψ s , ξ ∈ ⊗ ki =1 F ( H n i ) . Let Ψ : ( ⊗ ki =1 F ( H n i )) ⊗ C N → ⊗ ki =1 F ( H n i ) be the bounded operator having the 1 × N matrix repre-sentation [ e ψ ( R i,j ) , e ψ ( R i,j ) , . . . ] , where C ∞ stands for ℓ ( N ). Note that Ψ is a multi-analytic operator with respect to the universal model S and ΨΨ ∗ = P M . Therefore, Ψ is a partial isometry. Since S i,j are isometries, the initial space of Ψ,i.e. Ψ ∗ ( ⊗ ki =1 F ( H n i )) = { x ∈ ( ⊗ ki =1 F ( H n i )) ⊗ C N : k Ψ x k = k x k} is reducing under each operator S i,j ⊗ I C N . Consequently, since ∆ S ( I ) = P C , to prove that Ψ is anisometry, it is enough to show that C = P C e ψ s ( R i,j ) ∗ ( ⊗ ki =1 F ( H n i )) for each s ∈ { , . . . , N } . The latter equality is true since P C e ψ s ( R i,j ) ∗ ( ψ s ) = D e ψ s ( R i,j ) ∗ ( ψ s ) , E = k ψ s k = 1 . Therefore, Ψ is an isometry.To prove the converse of the theorem, assume that there is a sequence { ψ s } Ns =1 of right multipliers of ⊗ ki =1 F ( H n i ) such that { e ψ s ( R i,j ) } Ns =1 are isometries with orthogonal ranges and P M = N X s =1 e ψ s ( R i,j ) e ψ s ( R i,j ) ∗ . Since S i,j commutes with R r,q for any i, r ∈ { , . . . , k } , j ∈ { , . . . , n i } , and q ∈ { , . . . , n r } , and { e ψ s ( R i,j ) } Ns =1 are isometries with orthogonal ranges, we deduce that P M S i,j P M = S i,j P M . Therefore, M is an invariant subspace under the universal model S . Moreover, we have∆ M = ( id − Φ S ) ◦ · · · ◦ ( id − Φ S k )( P M ) = N X s =1 e ψ s ( R i,j ) ∆ S ( I ) e ψ s ( R i,j ) ∗ ≥ . Now, we prove that the sequence { ψ s } Ns =1 is uniquely determined up to a unitary equivalence. Let { ψ ′ s } N ′ s =1 be another sequence of right multipliers of ⊗ ki =1 F ( H n i ) such that { e ψ ′ s ( R i,j ) } N ′ s =1 are isometrieswith orthogonal ranges and P M = N ′ X s =1 e ψ ′ s ( R i,j ) e ψ ′ s ( R i,j ) ∗ . As above, we have ∆ M ξ = N X t =1 h ξ, ψ s i ψ s = N ′ X t =1 h ξ, ψ ′ s i ψ ′ s , ξ ∈ ⊗ ki =1 F ( H n i ) . Consequently, if ∆ M has finite rank then { ψ s } Ns =1 and { ψ ′ s } N ′ s =1 are orthonormal bases for the range of∆ M , thus N = N ′ ∈ N . If ∆ M does not have finite rank, similar arguments, show that N = N ′ = ∞ .Now, note that M = N M s =1 (cid:0) ⊗ ki =1 F ( H n i ) (cid:1) ψ s = N M s =1 (cid:0) ⊗ ki =1 F ( H n i ) (cid:1) ψ ′ s , and { ( e α ⊗ · · · ⊗ e kα k ) ψ s : α i ∈ F + n i , s ∈ { , . . . , N }} and { ( e α ⊗ · · · ⊗ e kα k ) ψ ′ s : α i ∈ F + n i , s ∈ { , . . . , N }} are orthonormal bases for M . Define the unitary operator U : M → M by setting U (( e α ⊗ · · · ⊗ e kα k ) ψ s ) := ( e α ⊗ · · · ⊗ e kα k ) ψ s . It is clear than U ( S i,j | M ) = ( S i,j | M ) U for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } . The proof is complete. (cid:3) If M is a Beurling type invariant subspace of S ⊗ I H , then ( S ⊗ I H ) | M := (( S ⊗ I H ) | M , . . . , ( S k ⊗ I H ) | M )is in the polyball B n ( M ), where ( S i ⊗ I H ) | M := (( S i, ⊗ I H ) | M , . . . , ( S i,n i ⊗ I H ) | M ). We say that M has finite rank if ( S ⊗ I H ) | M has finite rank. The next result shows that the Euler characteristiccompletely classifies the finite rank Beurling type invariant subspaces of S ⊗ I H which do not containreducing subspaces. In particular, the Euler characteristic classifies the finite rank Beurling type invariantsubspaces of F ( H n ) ⊗ · · · ⊗ F ( H n k ). Theorem 2.3.
Let M and N be invariant subspaces of ⊗ ki =1 F ( H n i ) ⊗ H which do not contain reducingsubspaces of S ⊗ I H and such that the defect operators ∆ M and ∆ N are positive and have finite ranks.Then M and N are unitarily equivalent if and only if χ (( S ⊗ I H ) | M ) = χ (( S ⊗ I H ) | N ) . ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 15
Proof.
Let M be a Beurling type invariant subspaces of ⊗ ki =1 F ( H n i ) ⊗ H . Then there is a Hilbertspace L and an isometric multi-analytic operator Ψ : ⊗ ki =1 F ( H n i ) ⊗ L → ⊗ ki =1 F ( H n i ) ⊗ H such that M = Ψ[ ⊗ ki =1 F ( H n i ) ⊗ L ]. Consequently, we have P M = ΨΨ ∗ and ∆ ( S ⊗ I ) | M ( I M ) = (cid:0) id − Φ ( S ⊗ I ) | M (cid:1) ◦ · · · ◦ (cid:0) id − Φ ( S k ⊗ I ) | M (cid:1) ( P M )= Ψ ( id − Φ S ⊗ I ) ◦ · · · ◦ ( id − Φ S k ⊗ I ) ( I )Ψ ∗ | M = Ψ( P C ⊗ I L )Ψ ∗ | M . If { ℓ ω } ω ∈ Ω is an orthonormal basis for L , then { v ω := Ψ(1 ⊗ ℓ ω ) : ω ∈ Ω } is an orthonormal set and { Ψ( e β ⊗ · · · ⊗ e kβ k ⊗ ℓ ω ) : β i ∈ F + n i , i ∈ { , . . . , k } , ω ∈ Ω } is an orthonormal basis for M . It is easy to see that Ψ( P C ⊗ I L )Ψ ∗ ( M ) coincides with the closure of therange of the defect operator ∆ ( S ⊗ I ) | M ( I M ) and also to the closed linear span of { v ω := Ψ(1 ⊗ ℓ ω ) : ω ∈ Ω } .This shows that rank [( S ⊗ I ) | M ] = card Ω = dim L . Assume that rank ( S ⊗ I ) | M ) = p = dim L . Taking into account that Ψ is a multi-analytic operatorand P M = ΨΨ ∗ , we deduce that (cid:16) id − Φ q +1( S ⊗ I ) | M (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1( S k ⊗ I ) | M (cid:17) ( I M ) = (cid:16) id − Φ q +1 S ⊗ I (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 S k ⊗ I (cid:17) ( P M )= Ψ (cid:16) id − Φ q +1 S ⊗ I L (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 S k ⊗ I L (cid:17) ( I )Ψ ∗ | M . Hence, we have rank h(cid:16) id − Φ q +1( S ⊗ I ) | M (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1( S k ⊗ I ) | M (cid:17) ( I M ) i = rank h(cid:16) id − Φ q +1 S ⊗ I L (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 S k ⊗ I L (cid:17) ( I ) i = rank h(cid:16) id − Φ q +1 S (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 S k (cid:17) ( I ) i dim L . Due to Theorem 1.2 and the fact that χ ( S ) = 1, we deduce that χ (( S ⊗ I ) | M ) = dim L = p = rank [( S ⊗ I ) | M ] . If M and N are finite rank Beurling type invariant subspaces of ⊗ ki =1 F ( H n i ) ⊗ H and ( S ⊗ I ) | M isunitarily equivalent to ( S ⊗ I ) | N , then χ (( S ⊗ I ) | M ) = χ (( S ⊗ I ) | N ) . To prove the converse, assume that thelatter equality holds. Due to the first part of the proof, we must have rank (( S ⊗ I ) | M ) = rank (( S ⊗ I ) | N ).This shows that the defect spaces associated with ( S ⊗ I ) | M and ( S ⊗ I ) | N have the same dimension.Now, the Wold decomposition from [22] implies that ( S ⊗ I ) | M is unitarily equivalent to ( S ⊗ I ) | N . Theproof is complete. (cid:3) Let M be an invariant subspace of the tensor product F ( H n ) ⊗ · · · ⊗ F ( H n k ) ⊗ E , where E is afinite dimensional Hilbert space. We introduce the Euler characteristic of of M ⊥ by setting χ ( M ⊥ ) := lim q →∞ · · · lim q k →∞ rank (cid:2) P M ⊥ ( P ≤ ( q ,...,q k ) ⊗ I E ) (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) . Theorem 2.4.
Let M be an invariant subspace of the tensor product F ( H n ) ⊗ · · · ⊗ F ( H n k ) ⊗ E ,where E is a finite dimensional Hilbert space. Then the Euler characteristic of M ⊥ exists and satisfiesthe equation χ ( M ⊥ ) = χ ( M ) , where M := ( M , . . . , M k ) with M i := ( M i, , . . . , M i,n i ) and M i,j := P M ⊥ ( S i,j ⊗ I E ) | M ⊥ .Proof. Taking into account that M is an invariant subspace under each operator S i,j ⊗ I E for i ∈{ , . . . , k } , j ∈ { , . . . , n i } , we deduce that M ∗ i,j M ∗ r,s = ( S ∗ i,j ⊗ I E )( S ∗ r,s ⊗ I E ) | M ⊥ . Consequently, we have ∆ pM ( I M ⊥ ) = P M ⊥ ∆ pS ⊗ I ( I ) | M ⊥ ≥ for any p = ( p , . . . , p k ) with p i ∈ { , } . Therefore, M is in the polyball B n ( M ⊥ ) and has finite rank.Since M ⊥ is invariant under S ∗ i,j ⊗ I E , we deduce thatrank h ( id − Φ q +1 M ) ◦ · · · ◦ ( id − Φ q k +1 M k )( I M ⊥ ) i = rank h P M ⊥ ( id − Φ q +1 S ⊗ I ) ◦ · · · ◦ ( id − Φ q k +1 S k ⊗ I )( I ) | M ⊥ i = rank (cid:2) P M ⊥ ( P ≤ ( q ,...,q k ) ⊗ I E ) | M ⊥ (cid:3) = rank (cid:2) P M ⊥ ( P ≤ ( q ,...,q k ) ⊗ I E ) (cid:3) for any q i ∈ Z + . Hence and using Theorem 1.2, we conclude that χ ( M ) exists and χ ( M ) = lim q →∞ · · · lim q k →∞ rank (cid:2) P M ⊥ ( P ≤ ( q ,...,q k ) ⊗ I E ) (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) = χ ( M ⊥ ) . This completes the proof. (cid:3)
In what follows, we prove a multiplicative property for the Euler characteristic.
Theorem 2.5.
Given a function κ : N → N and n ( i ) ∈ N κ ( i ) for i ∈ { , . . . , p } , let S ( n ( i ) ) and S ( n (1) ,..., n ( p ) ) be the universal models of the polyballs B n ( i ) and B ( n (1) ,..., n ( p ) ) , respectively. For each i ∈ { , . . . , p } ,assume that (i) E i is a finite dimensional Hilbert space; (ii) M i is an invariant subspace under S ( n ( i ) ) ⊗ I E i .Then the Euler characteristic satisfies the equation χ ( M ⊥ ⊗ · · · ⊗ M ⊥ p ) = p Y i =1 χ ( M ⊥ i ) , where, under the appropriate identification, M ⊥ ⊗ · · · ⊗ M ⊥ p is viewed as a coinvariant subspace for S ( n (1) ,..., n ( p ) ) ⊗ I E ⊗···⊗E p .Proof. For each i ∈ { , . . . , p } let n ( i ) := ( n ( i )1 , . . . , n ( i ) κ ( i ) ) ∈ N κ ( i ) . Given j ∈ { , . . . , κ ( i ) } , let F ( H ( i ) n j )be the full Fock space with n ( i ) j generators, and denote by P ( n ( i ) j ) q ( i ) j the orthogonal projection of F ( H n ( i ) j )onto the span of all homogeneous polynomials of F ( H n ( i ) j ) of degree equal to q ( i ) j ∈ Z + . Consider theorthogonal projections Q := P ( n (1)1 ) ≤ q (1)1 ⊗ · · · ⊗ P ( n (1) κ (1) ) ≤ q (1) κ (1) , · · · , Q p := P ( n ( p )1 ) ≤ q ( p )1 ⊗ · · · ⊗ P ( n ( p ) κ ( p ) ) ≤ q ( p ) κ ( p ) and let U be the unitary operator which provides the canonical identification of the Hilbert tensor product ⊗ pi =1 h F ( H ( i ) n ) ⊗ · · · ⊗ F ( H ( i ) n κ ( i ) ) ⊗ E i i with n ⊗ pi =1 h F ( H ( i ) n ) ⊗ · · · ⊗ F ( H ( i ) n κ ( i ) ) io ⊗ ( E ⊗ · · · ⊗ E p ).Note that U [( Q ⊗ I E ) ⊗ · · · ⊗ ( Q p ⊗ I E p )] = [ Q ⊗ · · · ⊗ Q p ⊗ I E ⊗···⊗E p ] U and U ( M ⊥ ⊗ · · · ⊗ M ⊥ p ) is a coinvariant subspace under S ( n (1) ,..., n ( p ) ) ⊗ I E ⊗···⊗E p . Due to Theorem 2.4,we obtain χ (cid:0) ⊗ ki =1 M ⊥ i (cid:1) = lim rank h P U ( M ⊥ ⊗···⊗M ⊥ p ) (cid:0) Q ⊗ · · · ⊗ Q p ⊗ I E ⊗···⊗E p (cid:1)i rank [ Q ⊗ · · · ⊗ Q p ]= lim rank h U ∗ P U ( M ⊥ ⊗···⊗M ⊥ p ) U U ∗ (cid:0) Q ⊗ · · · ⊗ Q p ⊗ I E ⊗···⊗E p (cid:1) U i rank [ Q ⊗ · · · ⊗ Q p ]= lim rank n P M ⊥ ⊗···⊗M ⊥ p (cid:2) ( Q ⊗ I E ) ⊗ · · · ⊗ (cid:0) Q p ⊗ I E p (cid:1)(cid:3)o rank [ Q ⊗ · · · ⊗ Q p ] , ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 17 where the limit is taken over ( q (1)1 , . . . , q (1) κ (1) , . . . , q ( p )1 , . . . , q ( p ) κ ( p ) ) ∈ Z κ (1)+ ··· + κ ( p )+ . We remark that the latterlimit is equal to the productlim ( q (1)1 ,...,q (1) κ (1) ) ∈ Z κ (1)+ rank h P M ⊥ ( Q ⊗ I E ) i rank [ Q ] · · · lim ( q ( p )1 ,...,q ( p ) κ ( p ) ) ∈ Z κ ( p )+ rank h P M ⊥ p (cid:0) Q p ⊗ I E p (cid:1)i rank [ Q p ] , which, due to Theorem 2.4, is equal to Q pi =1 χ (cid:0) M ⊥ i (cid:1) . Therefore, we have χ (cid:0) ⊗ pi =1 M ⊥ i (cid:1) = p Y i =1 χ (cid:0) M ⊥ i (cid:1) . The proof is complete. (cid:3)
As a particular case, we remark that if M ⊥ ⊂ H ( D n ) ⊗ C r and N ⊥ ⊂ H ( D p ) ⊗ C q are coinvariantsubspaces, then so is M ⊥ ⊗ N ⊥ ⊂ H ( D n + p ) ⊗ C rq and the Euler characteristic has the multiplicativeproperty χ ( M ⊥ ⊗ N ⊥ ) = χ ( M ⊥ ) χ ( N ⊥ ).The next result shows that there are many non-isomorphic pure elements in the polyball with the sameEuler characteristic. Theorem 2.6.
Let n = ( n , . . . , n k ) ∈ N k be such that k ≥ and n i ≥ for all i ∈ { , . . . , k } . Then, foreach t ∈ (0 , , there exists an uncountable family { T ( ω ) ( t ) } ω ∈ Ω of pure elements in the regular polyball B n with the following properties: (i) T ( ω ) ( t ) in not unitarily equivalent to T ( σ ) ( t ) for any ω, σ ∈ Ω , ω = σ . (ii) rank [ T ( ω ) ( t )] = 1 and χ [ T ( ω ) ( t )] = t for all ω ∈ Ω . Proof. If t ∈ [0 , { k p } Np =1 , 1 ≤ k < k < · · · , where N ∈ N or N = ∞ , and d p ∈ { , , . . . , n i − } , such that 1 − t = P Np =1 d p n kpi . Define the following subsetsof F + n i , the unital free semigroup on n i generators g i , . . . , g in i and the identity g i : J i := (cid:8) ( g i ) k , . . . , ( g id ) k (cid:9) ,J p := n ( g i ) k p − k p − ( g n i ) k p − , ( g i ) k p − k p − ( g in i ) k p − , . . . , ( g id p ) k p − k p − ( g in i ) k p − o , p = 2 , , . . . , N. We remark that(2.1) M i ( t ) := M β ∈∪ Np =1 J ip F ( H n i ) ⊗ e iβ is an invariant subspace of F ( H n i ). Let k p ≤ q i < k p +1 and note thatrank h P M i ( t ) ⊥ P ( i ) q i i rank h P ( i ) q i i = 1rank h P ( i ) q i i X β i ∈ F + ni D P M i ( t ) ⊥ P ( i ) q i e iβ i , e iβ i E = 1 − X β i ∈ F + ni D P ( i ) q i P M i ( t ) P ( i ) q i e iβ i , e iβ i E n q i i = 1 − n q i i X β i ∈ F + ni , | β i | = q i k P M i ( t ) e iβ i k = 1 − d n q i − k i + · · · + d p n q i − k p i n q i i = 1 − d n k i + · · · + d p n k p i ! . Consequently, we have lim q i →∞ rank h P M i ( t ) ⊥ P ( i ) q i i rank h P ( i ) q i i = 1 − N X p =1 d p n k p = t. Now, using Theorem 2.4 (when k = 1) and the Stoltz-Cesaro limit theorem, we deduce that χ i [ P M i ( t ) ⊥ S i | M i ( t ) ⊥ ] = lim q i →∞ rank h P M i ( t ) ⊥ P ( i ) ≤ q i i rank h P ( i ) ≤ q i i = lim q i →∞ P q i s i =0 rank h P M i ( t ) ⊥ P ( i ) s i iP q i s i =0 rank h P ( i ) s i i = lim q i →∞ rank h P M i ( t ) ⊥ P ( i ) q i i rank h P ( i ) q i i = 1 − N X p =1 d p n k p = t, where χ i stands for the Euler characteristic on the polyball B n i . Let t ∈ (0 ,
1) and ω ∈ ( t, M ( ω ) ( t ) := M ( ω ) ⊥ ⊗ M (cid:18) tω (cid:19) ⊥ ⊗ F ( H n ) ⊗ · · · ⊗ F ( H n k ) ! ⊥ . Note that M ( ω ) ( t ) is an invariant subspace under S i,j for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } . Setting T ( ω ) ( t ) := P M ( ω ) ( t ) ⊥ S | M ( ω ) ( t ) ⊥ and using Theorem 2.5, we deduce that χ [ T ( ω ) ( t )] = χ (cid:0) P M ( ω ) ⊥ S | M ( ω ) ⊥ (cid:1) χ (cid:16) P M ( tω ) ⊥ S | M ( tω ) ⊥ (cid:17) = t. Let σ ∈ (0 , t ). Since the Euler characteristic is a unitary invariant and χ [ P M ( ω ) ⊥ S | M ( ω ) ⊥ ] = ω = σ = χ [ P M ( σ ) ⊥ S | M ( σ ) ⊥ ] , we deduce that P M ( ω ) ⊥ S | M ( ω ) ⊥ is not unitary equivalent to P M ( σ ) ⊥ S | M ( σ ) ⊥ . According to Theorem2.10 from [24], when E = C , we have M ( ω ) = M ( σ ), which implies M ( ω ) ( t ) = M ( σ ) ( t ). Using againthe above-mentioned result from [24], we conclude that T ( ω ) ( t ) in not unitarily equivalent to T ( σ ) ( t ) forany ω, σ ∈ (0 , t ), ω = σ . The fact that rank [ T ( ω ) ( t )] = 1 is obvious. The proof is complete. (cid:3) Corollary 2.7.
Let n = ( n , . . . , n k ) ∈ N k be such that n i ≥ and let t ∈ [0 , . Then there exists a pureelement T in the polyball B n such that rank ( T ) = 1 and χ ( T ) = t. Proof. If t = 1, the Euler characteristic χ ( S ) = 1. When t ∈ [0 , M ( t ) :=( M ( t ) ⊥ ⊗ F ( H n ) ⊗ · · · ⊗ F ( H n k )) ⊥ , where M ( t ) is defined by relation (2.1). As in the proof ofTheorem 2.6, one can show that χ ( P M ( t ) ⊥ S | M ( t ) ⊥ ) = t . This completes the proof. (cid:3) Using Corollary 2.7 and tensoring with the identity on C m , one can easily see that for any t ∈ [0 , m ],there exists a pure element T in the polyball B n , with rank ( T ) = m and χ ( T ) = t . Consequently, therange of the Euler characteristic coincides with the interval [0 , ∞ ). Theorem 2.8.
Let T ∈ B n ( H ) have finite rank and let M be an invariant subspace under T such that T | M ∈ B n ( M ) and dim M ⊥ < ∞ . Then T | M has finite rank and | χ ( T ) − χ ( T | M ) | ≤ dim M ⊥ k Y i =1 ( n i − . Proof.
First, note that rank ( T | M ) = rank ∆ T ( P M ). Taking into account the fact that ∆ T ( P M ) = ∆ T ( I H ) − ∆ T ( P M ⊥ ), we deduce that rank ( T | M ) < ∞ . On the other hand, since M is an invariant ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 19 subspace under T , we haverank h(cid:16) id − Φ q +1 T | M (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k | M (cid:17) ( I M ) i = rank h(cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( P M ) i ≤ rank h(cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( I H ) i + rank h(cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( P M ⊥ ) i . Since rank h(cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( P M ⊥ ) i ≤ (1 + n q +11 ) · · · (1 + n q k +1 k )rank [ P M ⊥ ] , we deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) rank h(cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( I H ) iQ ki =1 (1 + n i + · · · + n q i i ) − rank h(cid:16) id − Φ q +1 T | M (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k | M (cid:17) ( I M ) iQ ki =1 (1 + n i + · · · + n q i i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 + n q +11 ) · · · (1 + n q k +1 k ) Q ki =1 (1 + n i + · · · + n q i i ) trace [ P M ⊥ ] . Since n i ≥
2, we can use Theorem 1.2 and obtain | χ ( T ) − χ ( T | M ) | ≤ dim M ⊥ k Y i =1 ( n i − . The proof is complete. (cid:3)
Theorem 2.9.
Let T ∈ B n ( H ) have finite rank and let M be a co-invariant subspace under T with dim M ⊥ < ∞ . Then P M T | M has finite rank and χ ( T ) = χ ( P M T | M ) . Proof.
Denote A := ( A , . . . , A k ) and A i := ( A i, , . . . , A i,n i ), where A i,j := P M T i,j | M for i ∈ { , . . . , k } and j ∈ { , . . . , n i } . Using the fact that Φ q i A i ( I M ) = P M Φ q i T i ( I H ) | M , we deduce thatrank h(cid:16) id − Φ q +1 A (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 A k (cid:17) ( I M ) i ≤ rank h(cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( I H ) i ≤ rank h P M (cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( I H ) i + rank h P M ⊥ (cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( I H ) i ≤ rank h(cid:16) id − Φ q +1 A (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 A k (cid:17) ( I M ) i + rank h P M ⊥ (cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( I H ) i . Note also that rank h P M ⊥ (cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( I H ) i ≤ dim M ⊥ . Hence, we deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) rank h(cid:16) id − Φ q +1 T (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 T k (cid:17) ( I H ) iQ ki =1 (1 + n i + · · · + n q i i ) − rank h(cid:16) id − Φ q +1 A (cid:17) ◦ · · · ◦ (cid:16) id − Φ q k +1 A k (cid:17) ( I M ) iQ ki =1 (1 + n i + · · · + n q i i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Q ki =1 (1 + n i + · · · + n q i i ) dim M ⊥ . Now, using Theorem 1.2, we conclude that χ ( T ) = χ ( A ), which completes the proof. (cid:3) According to [18], there are invariant subspaces M ⊂ F ( H n ) which do not contain nonzero poly-nomials in F ( H n ). This implies that the set { P M ⊥ e α } α ∈ F + n , | α |≤ m is linearly independent for any m ∈ N . Consequently, the set { ( P M ⊥ ⊗ I ⊗ · · · ⊗ I )( e α ⊗ · · · ⊗ e kα k ) } α i ∈ F + ni , | α i |≤ m is linearly independentfor each m ∈ N . This shows that the subspace M ⊗ F ( H n ) ⊗ · · · ⊗ F ( H n k ) does not contain nonzero polynomials. On the other hand, according to [15] (see also [16]), any invariant subspace M ⊂ F ( H n )is of Beurling type. Therefore, M ⊗ F ( H n ) ⊗ · · · ⊗ F ( H n k ) is also of Beurling type. Lemma 2.10.
Let
M ⊂ F ( H n ) ⊗ · · · ⊗ F ( H n k ) be an nonzero invariant subspace satisfying either oneof the following conditions: (i) M is a Beurling type invariant subspace which does not contain any polynomial; (ii) M = ( M ⊥ ⊗ · · · M ⊥ k ) ⊥ , where { } 6 = M i ⊂ F ( H n i ) are invariant subspaces which do notcontain nonzero polynomials.Then curv ( P M ⊥ S | M ⊥ ) < and χ ( P M ⊥ S | M ⊥ ) = 1 . Proof.
Assume that
M ⊂ F ( H n ) ⊗ · · · ⊗ F ( H n k ) is an nonzero invariant subspace which does notcontain any polynomial. Then the set { P M ⊥ ( e α ⊗ · · · ⊗ e kα k ) } α i ∈ F + ni , | α i |≤ m is linearly independent foreach m ∈ N and, consequently, using Theorem 2.4, we deduce that χ ( P M ⊥ S | M ⊥ ) = 1 . If, in addition, M is a Beurling type invariant subspace, then due to [24] (see Proposition 3.14), we havecurv ( P M ⊥ S | M ⊥ ) < M = ( M ⊥ ⊗ · · · M ⊥ k ) ⊥ , where { } 6 = M i ⊂ F ( H n i ) are invariantsubspaces which do not contain nonzero polynomials. Due to the first part of the proof, when k = 1, wehave χ ( P M ⊥ i S i | M ⊥ i ) = 1 and curv ( P M ⊥ i S i | M ⊥ i ) <
1. Due to Theorem 2.4 and Theorem 2.5, we have χ ( P M ⊥ S | M ⊥ ) = χ ( M ⊥ ⊗ · · · M ⊥ k ) = k Y i =1 χ ( P M ⊥ i S i | M ⊥ i ) = 1 . On the other hand, using [24] (see Corollary 2.5), we obtaincurv ( P M ⊥ S | M ⊥ ) = curv ( P M ⊥ ⊗···M ⊥ k S | M ⊥ ⊗···M ⊥ k ) = k Y i =1 curv ( P M ⊥ i S | M ⊥ i ) < (cid:3) The next result shows that the curvature and the Euler characteristic can be far from each each other.
Proposition 2.11.
For every ǫ ∈ (0 , and m ∈ N , there exists k ∈ N and T ∈ B n ( H ) with n =( n , . . . , n k ) and n i ≥ such that < curv ( T ) < ǫ and χ ( T ) = rank ( T ) = m. Proof.
Due to Lemma 2.10, there is an invariant subspace M ⊆ F ( H n ) such that χ ( P M ⊥ S | M ⊥ ) = 1and t := curv ( P M ⊥ S | M ⊥ ) <
1. Given ǫ >
0, take k ∈ N such that t k < ǫ . Consider n = · · · = n k andlet M := ( M ⊥ ⊗ · · · M ⊥ ) ⊥ . As in the proof of Lemma 2.10, we havecurv ( P M ⊥ S | M ⊥ ) = t k < ǫ and χ ( P M ⊥ S | M ⊥ ) = 1 . The proof is complete. (cid:3)
ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 21 A Gauss-Bonnet-Chern type theorem on noncommutative polyballs
In this section, we provide a characterization of the graded invariant subspaces for the tensor product F ( H n ) ⊗ · · · ⊗ F ( H n k ), and obtain a version of the Gauss-Bonnet-Chern theorem on noncommutativepolyballs.A Hilbert space H is called Z k -graded if there is a strongly continuous unitary representation U : T k → B ( H ) of the k -torus T k = T × · · · × T , where T := { z ∈ C : | z | = 1 } . U is called the gauge group of H . Definition 3.1.
A tuple T = ( T , . . . , T k ) ∈ B ( H ) n × · · · × B ( H ) n k with T i = ( T i, , . . . , T i,n i ) , is called Z k -graded if there is a distinguished gauge group U on H such that U ( λ ) T i,j U ( λ ) ∗ = λ i T i,j , λ = ( λ , . . . , λ k ) ∈ T k , for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } . A closed subspace
M ⊆ H is called graded with respect to U if U ( λ ) M ⊆ M for all λ ∈ T k . In thiscase, the subrepresentation U | M is the gauge group on M . Similarly, the orthocomplement M ⊥ := H⊖M is called graded if U ( λ ) M ⊥ ⊆ M ⊥ for all λ ∈ T k . Note that M is graded if and only if M ⊥ is graded.Assume that T is graded with respect to the gauge group U on H and let M ⊆ H be an invariantsubspace under T . Note that if M is graded with respect to U , then T | M is graded with respect to thesubrepresentation U | M , i.e. U ( λ ) | M ( T i,j | M ) = λ i ( T i,j | M ) U ( λ ) | M , λ = ( λ , . . . , λ k ) ∈ T k , for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } . Similarly, M ⊥ is graded with respect to U and P M ⊥ T | M ⊥ isgraded with respect to the subrepresentation U | M ⊥ .We remark that the universal model S = ( S , . . . , S k ), S i = ( S i, , . . . , S i,j ), with S i,j acting onthe tensor product F ( H n ) ⊗ · · · ⊗ F ( H n k ), is Z k -graded with respect to the canonical gauge group U : T k → B ( F ( H n ) ⊗ · · · ⊗ F ( H n k )) defined by U ( λ ) X αi ∈ F + nii ∈{ ,...,k } a α ,...,α k e α ⊗ · · · ⊗ e kα k = X αi ∈ F + nii ∈{ ,...,k } λ | α | · · · λ | α | k a α ,...,α k e α ⊗ · · · ⊗ e kα k . Given ( s , . . . , s k ) ∈ Z k + , we say that a polynomial in F ( H n ) ⊗ · · · ⊗ F ( H n k ) is multi-homogeneous oforder ( s , . . . , s k ) if it has the form q = X αi ∈ F + ni , | αi | = sii ∈{ ,...,k } a α ,...,α k e α ⊗ · · · ⊗ e kα k . This is equivalent to the fact that U ( λ ) q = λ s · · · λ s k k q for any λ = ( λ , . . . , λ k ) ∈ T k .Let M ⊂ F ( H n ) ⊗ · · · ⊗ F ( H n k ) be an invariant subspace for the universal model S . We recall (seeCorollary 2.2 from [24]) that the defect map ∆ S is one-to-one and any projection P M has the followingTaylor type representation around its defect P M = ∞ X s =0 Φ s S ∞ X s =0 Φ s S · · · ∞ X s k =0 Φ s k S k ( ∆ S ( P M )) · · · !! , where the iterated series converge in the weak operator topology. If, in addition, ∆ S ( P M ) ≥
0, then(3.1) P M = X ( s ,...,s k ) ∈ Z k + Φ s S ◦ · · · ◦ Φ s k S k ( ∆ S ( P M )) . We recall that defect operator of M is given by ∆ M := ∆ S ( P M ). Note that ∆ M ( M ) ⊆ M and∆ M ( M ⊥ ) = { } . Moreover, we have ∆ S | M ( I M ) = ∆ M | M . Now, it is clear that ∆ M is positive and hasfinite rank if and only if ∆ S | M ( I M ) has the same properties. The remarks above show that the invariantsubspace M is uniquely determined by its defect operator ∆ M . Note that any set { ψ s } s ∈ Λ , Λ ⊆ N , of multi-homogeneous polynomials (perhaps of different orders)generates a graded closed invariant subspace of F ( H n ) ⊗ · · · ⊗ F ( H n k ) by setting M := span { S ,α · · · S k,α k ψ s : α ∈ F + n i , i ∈ { , . . . , k } , s ∈ Λ } . The next theorem provides a characterization of those graded invariant subspaces
M ⊆ F ( H n ) ⊗ · · · ⊗ F ( H n k ) with positive defect operator ∆ M .If f := P β i ∈ F + ni a β ,...,β k e β ⊗· · ·⊗ e kβ k is a polynomial in F ( H n ) ⊗· · ·⊗ F ( H n k ), and X i,j are boundedoperators on a Hilbert space, where i ∈ { , . . . , k } and j ∈ { , . . . , n i } , we define the polynomial calculus f ( X i,j ) := P β i ∈ F + ni a β ,...,β k X ,β · · · X k,β k . We use the notation e f ( R i,j ) := P β i ∈ F + ni a β ,...,β k R , e β · · · R k, e β k ,where e β i = g ij p · · · g ij denotes the reverse of β i = g ij · · · g ij p ∈ F + n i . Theorem 3.2.
Let
M ⊆ F ( H n ) ⊗ · · · ⊗ F ( H n k ) be an invariant subspace. Then M is graded and haspositive defect operator ∆ M if and only if there is a sequence of multi-homogeneous polynomials { ψ s } Ns =1 ,where N ∈ N or N = ∞ , with the following properties: (i) each ψ s is in the range of ∆ M ; (ii) { e ψ s ( R i,j ) } Ns =1 are isometries with orthogonal ranges, where { R i,j } is the right universal model; (iii) The orthogonal projection P M satisfies the relation P M = N X s =1 e ψ s ( R i,j ) e ψ s ( R i,j ) ∗ , where the convergence is in the strong operator topology.Moreover, in this case, we have N = rank (∆ M ) and ∆ M ξ = N X t =1 h ξ, ψ s i ψ s , ξ ∈ ⊗ ki =1 F ( H n i ) . Proof.
Assume that
M ⊆ F ( H n ) ⊗ · · · ⊗ F ( H n k ) is a graded invariant subspace and has positivedefect operator ∆ M . Let U : T k → B ( F ( H n ) ⊗ · · · ⊗ F ( H n k )) be the canonical gauge group of F ( H n ) ⊗ · · · ⊗ F ( H n k ). Then U ( λ , . . . , λ k ) M = M for any ( λ , . . . , λ k ) ∈ T k and, consequently U ( λ , . . . , λ k ) commutes with P M . Taking into account that U ( λ , . . . , λ k ) S i,j U ( λ , . . . , λ k ) ∗ = λ i S i,j , λ = ( λ , . . . , λ k ) ∈ T k , for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } , one can easily see that U ( λ , . . . , λ k )∆ M = ∆ M U ( λ , . . . , λ k ) , λ = ( λ , . . . , λ k ) ∈ T k . For each ( s , . . . , s k ) ∈ Z k + , let Q ( s ,...,s k ) be the orthogonal projection of F ( H n ) ⊗ · · · ⊗ F ( H n k ) ontothe subspace of multi-homogeneous polynomials of order ( s , . . . , s k ). Since U ( λ , . . . , λ k ) = ∞ X p =0 · · · ∞ X p k =0 λ α · · · λ αk Q ( s ,...,s k ) , λ = ( λ , . . . , λ k ) ∈ T k , and due to the spectral theorem, we deduce that∆ M Q ( s ,...,s k ) = Q ( s ,...,s k ) ∆ M , ( s , . . . , s k ) ∈ Z k . If ( p , . . . , p k ) is in the set Ω := { ( s , . . . , s k ) ∈ Z k : ∆ M Q ( s ,...,s k ) = 0 } , then ∆ M Q ( p ,...,p k ) is a nonzero positive finite rank operator supported in the space of multi-homogeneouspolynomials Q ( p ,...,p k ) ( ⊗ ki =1 F ( H n i )). Due to the spectral theorem, it can be expressed as a finite sum∆ M Q ( p ,...,p k ) = P mt =1 Λ t of rank-one positive operators Λ t defined by(3.2) Λ t ξ := h ξ, ψ t i ψ t , ξ ∈ ⊗ ki =1 F ( H n i ) , ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 23 where { ψ t } mt =1 are orthonormal multi-homogeneous polynomials in ∆ M Q ( p ,...,p k ) ( ⊗ ki =1 F ( H n i )). Notethat e ψ t ( R i,j ) P C e ψ t ( R i,j ) ∗ ξ = e ψ t ( R i,j )( h ξ, ψ t i ) = h ξ, ψ t i ψ t = Λ t ξ. Consequently, using the fact that e ψ t ( R i,j ) is a multi-analytic operator with respect to the universal model S , we have(3.3) q X s =0 · · · q k X s k =0 Φ s S ◦ · · · ◦ Φ s k S k (Λ t ) = e ψ t ( R i,j ) q X s =0 · · · q k X s k =0 Φ s S ◦ · · · ◦ Φ s k S k ( P C ) ! e ψ t ( R i,j ) ∗ Taking the limits as q → ∞ , . . . , q k → ∞ , and using the identity I = X ( s ,...,s k ) ∈ Z k + Φ s S ◦ · · · ◦ Φ s k S k ( ∆ S ( I )) = X ( s ,...,s k ) ∈ Z k + Φ s S ◦ · · · ◦ Φ s k S k ( P C ) , we deduce that(3.4) ∞ X s =0 · · · ∞ X s k =0 Φ s S ◦ · · · ◦ Φ s k S k (Λ t ) = e ψ t ( R i,j ) e ψ t ( R i,j ) ∗ . Now, let { Λ t } Nt =1 , where N ∈ N or N = ∞ , be the set of the rank-one positive operators associated, asabove, with all the operators ∆ M Q ( p ,...,p k ) with ( p , . . . , p k ) ∈ Ω, and let { ψ t } Nt =1 be the correspondingorthonormal multi-homogeneous polynomials, according to relation (3.2). We have ∆ M = P Nt =1 Λ t wherethe convergence in the strong operator topology, and rank ∆ M = N . Note also that∆ M ξ = N X t =1 h ξ, ψ t i ψ t , ξ ∈ ⊗ ki =1 F ( H n i ) . Due to the fact each completely positive map Φ S i is W OT -continuous and using relation (3.3), we deducethat that q X s =0 · · · q k X s k =0 Φ s S ◦ · · · ◦ Φ s k S k (∆ M ) = ∞ X t =1 q X s =0 · · · q k X s k =0 Φ s S ◦ · · · ◦ Φ s k S k (Λ t ) ! = ∞ X t =1 e ψ t ( R i,j ) q X s =0 · · · q k X s k =0 Φ s S ◦ · · · ◦ Φ s k S k ( P C ) ! e ψ t ( R i,j ) ∗ . Due to relation (3.1), { P q s =0 · · · P q k s k =0 Φ s S ◦ · · · ◦ Φ s k S k (∆ M ) } is an increasing sequence of positive opera-tors which converges strongly to P M as q → ∞ , . . . , q k → ∞ , and { P q s =0 · · · P q k s k =0 Φ s S ◦ · · · ◦ Φ s k S k ( P C ) } is an increasing sequence of positive operators which converges strongly to the identity operator on ⊗ ki =1 F ( H n i ). Consequently, the equalities above and relation (3.3) can be used to deduce that P M = ∞ X t =1 e ψ t ( R i,j ) e ψ t ( R i,j ) ∗ , where the convergence is in the strong operator topology. Let Ψ : ( ⊗ ki =1 F ( H n i )) ⊗ C N → ⊗ ki =1 F ( H n i )be the bounded operator having the 1 × N matrix representation[ e ψ ( R i,j ) , e ψ ( R i,j ) , . . . ] , where C ∞ stands for ℓ ( N ). Note that Ψ is a multi-analytic operator with respect to the universal model S and ΨΨ ∗ = P M . Therefore, Ψ is a partial isometry. Since S i,j are isometries, the initial space of Ψ,i.e. Ψ ∗ ( ⊗ ki =1 F ( H n i )) = { x ∈ ( ⊗ ki =1 F ( H n i )) ⊗ C N : k Ψ x k = k x k} is reducing under each operator S i,j ⊗ I C N . Consequently, since ∆ S ( I ) = P C , to prove that Ψ is anisometry, it is enough to show that C = P C e ψ t ( R i,j ) ∗ ( ⊗ ki =1 F ( H n i )) for each t ∈ { , . . . , N } . The latter equality is true due to the fact that P C e ψ t ( R i,j ) ∗ ( g t ) = D e ψ t ( R i,j ) ∗ ( ψ t ) , E = k ψ t k = 1 . The converse of the theorem is straightforward. (cid:3)
The next result is a Gauss-Bonnet-Chern type theorem for rank-one, graded, and pure elements innoncommutative polyballs.
Theorem 3.3. If T is a graded pure element of rank one in the noncommutative polyball B n ( H ) , with n i ≥ , then curv ( T ) = χ ( T ) . Proof.
Let U : T k → B ( H ) be a strongly continuous unitary representation of the k -torus and assumethat T is graded with respect to the gauge group U . The spectral subspaces of U are H s = H ( s ,...,s k ) := (cid:8) h ∈ H : U ( λ , . . . , λ k ) h = λ s · · · λ s k k h for ( λ . . . , λ k ) ∈ T k (cid:9) , where s = ( s , . . . , s k ) ∈ Z k . With respect to the orthogonal decomposition H = L s ∈ Z k H s we have U ( λ , . . . , λ k ) = X ( s ,...,s k ) ∈ Z k λ s · · · λ s k k Q ( s ,...,s k ) , ( λ . . . , λ k ) ∈ T k , where Q ( s ,...,s k ) is the orthogonal projection of H onto H ( s ,...,s k ) . Note that if T is graded with respectto the gauge group U , then T i,j H ( s ,...,s k ) ⊆ H ( s ,...,s i − ,s i +1 ,s i +1 ,...,s k ) , ( s , . . . , s k ) ∈ Z k , for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } . Indeed, if h ∈ H ( s ,...,s k ) , then U ( λ , . . . , λ k ) h = λ s · · · λ s k k h and U ( λ , . . . , λ k ) T i,j h = λ i T i,j U ( λ , . . . , λ k ) h = λ s · · · λ s i − i − λ s i +1 i λ s i +1 i +1 · · · λ s k k T i,j h for any ( λ . . . , λ k ) ∈ T k . Consequently, T i,j h ∈ H ( s ,...,s i − ,s i +1 ,s i +1 ,...,s k ) , which proves our assertion.Therefore,(3.5) T i,j Q ( s ,...,s k ) = Q ( s ,...,s i − ,s i +1 ,s i +1 ,...,s k ) T i,j for any ( s , . . . , s k ) ∈ Z k , i ∈ { , . . . , k } , and j ∈ { , . . . , n i } . This also implies(3.6) Q ( s ,...,s k ) T ∗ i,j = T ∗ i,j Q ( s ,...,s i − ,s i +1 ,s i +1 ,...,s k ) . Consequently, taking into account that U ( λ , . . . , λ k ) Q ( s ,...,s k ) = λ s · · · λ s k k Q ( s ,...,s k ) for any ( λ , . . . , λ k ) ∈ T k , one can easily check that ∆ T ( I ) U ( λ , . . . , λ k ) Q ( s ,...,s k ) = U ( λ , . . . , λ k ) Q ( s ,...,s k ) ∆ T ( I )for any ( s , . . . , s k ) ∈ Z k , which implies ∆ T ( I ) / U ( λ , . . . , λ k ) = U ( λ , . . . , λ k ) ∆ T ( I ) / , ( λ , . . . , λ k ) ∈ T k . Due to the spectral theorem, we deduce that ∆ T ( I ) / Q ( s ,...,s k ) = Q ( s ,...,s k ) ∆ T ( I ) / , ( s , . . . , s k ) ∈ Z k . Now, we assume that T has rank one. Then there exists ( d , . . . , d k ) ∈ Z k + such that(3.7) ∆ T ( I ) / = Q ( d ,...,d k ) ∆ T ( I ) / = ∆ T ( I ) / Q ( d ,...,d k ) = 0and ∆ T ( I ) / Q ( s ,...,s k ) = 0 for any ( s , . . . , s k ) ∈ Z k with ( s , . . . , s k ) = ( d , . . . , d k ). Consequently, ∆ T ( I ) / ( H ) = ∆ T ( I ) / H ( d ,...,d k ) ⊆ M s ≥ d ,...,s k ≥ d k H ( s ,...,s k ) . ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 25
Since L s ≥ d ,...,s k ≥ d k H ( s ,...,s k ) is invariant under each operator T i,j for any i ∈ { , . . . , k } and j ∈{ , . . . , n i } , we have(3.8) T ,β · · · T k,β k ∆ T ( I ) / H ⊆ M s ≥ d ,...,s k ≥ d k H ( s ,...,s k ) for any β i ∈ F + n i , i = 1 , . . . , k . On the other hand, the noncommutative Berezin kernel K T : H → F ( H n ) ⊗ · · · ⊗ F ( H n k ) ⊗ ∆ T ( I )( H )satisfies the relation(3.9) K ∗ T ( e β ⊗ · · · ⊗ e kβ k ⊗ h ) = T ,β · · · T k,β k ∆ T ( I ) / h for any β i ∈ F + n i , i = 1 , . . . , k , and h ∈ H . Since T is a pure element in the polyball, the Berezin kernel K T is an isometry and, therefore, H = range K ∗ T = span { T ,β · · · T k,β k ∆ T ( I ) / h : β i ∈ F + n i , h ∈ H} . Hence, and using relation (3.8), we deduce that H = M s ≥ d ,...,s k ≥ d k H ( s ,...,s k ) . On the other hand, taking into account relations (3.5), (3.6), and (3.7), we have ∆ T ( I ) / T ∗ k,α k · · · T ∗ ,α k T ,β · · · T k,β k ∆ T ( I ) / Q ( d ,...,d k ) = ∆ T ( I ) / T ∗ k,α k · · · T ∗ ,α k Q ( d + | β | ,...,d k + | β k | ) T ,β · · · T k,β k ∆ T ( I ) / = ∆ T ( I ) / Q ( d + | β |−| α | ,...,d k + | β k |−| α k | ) T ∗ k,α k · · · T ∗ ,α k T ,β · · · T k,β k ∆ T ( I ) / = ( Q ( d ,...,d k ) ∆ T ( I ) / T ∗ k,α k · · · T ∗ ,α k T ,β · · · T k,β k ∆ T ( I ) / if | β | = | α | , . . . , | β k | = | α k | α , β ∈ F + n , . . . , α k , β k ∈ F + n k . Consequently, we have(3.10) ∆ T ( I ) / T ∗ k,α k · · · T ∗ ,α k T ,β · · · T k,β k ∆ T ( I ) / = 0for any α , β ∈ F + n , . . . , α k , β k ∈ F + n k , except when | β | = | α | , . . . , | β k | = | α k | . Note that due to relations(3.9) and (3.10), for any ( q , . . . , q k ) ∈ Z k + , we have( P (1) q ⊗ · · · ⊗ P ( k ) q k ⊗ I H ) K T K ∗ T ( e β ⊗ · · · ⊗ e kβ k ⊗ h )= X α ∈ F + n , | α | = q · · · X α k ∈ F + nk , | α k | = q k e α ⊗ · · · ⊗ e kα k ⊗ ∆ T ( I ) / T ∗ k,α k · · · T ∗ ,α k T ,β · · · T k,β k ∆ T ( I ) / h if | β | = q , . . . , | β k | = q k , and zero otherwise. Now, one can see that K T K ∗ T ( P (1) q ⊗ · · · ⊗ P ( k ) q k ⊗ I H ) = ( P (1) q ⊗ · · · ⊗ P ( k ) q k ⊗ I H ) K T K ∗ T . for any ( q , . . . , q k ) ∈ Z k + , where P ( i ) q i is the orthogonal projection of the full Fock space F ( H n i ) ontothe span of all vectors e iα i with α ∈ F + n i and | α i | = q i . Since K T K ∗ T is a projection, we deduce that K T K ∗ T ( P (1) q ⊗ · · · ⊗ P ( k ) q k ⊗ I H ) } ( q ,...,q k ) ∈ Z k + is a net of orthogonal projections with orthogonal ranges.This implies that X ( s ,...,sk ) ∈ Z k +( s ,...,sk ) ≤ ( q ,...,qk ) K T K ∗ T ( P (1) q ⊗ · · · ⊗ P ( k ) q k ⊗ I H ) = K T K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I )is an orthogonal projection for any ( q , . . . , q k ) ∈ Z k + . Consequently, we havetrace [ K T K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I )] = rank [ K T K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I )] . Applying Theorem 1.2 and [24] (see Theorem 1.3), and using the fact that K T is an isometry, we deducethatcurv ( T ) = lim q →∞ · · · lim q k →∞ trace (cid:2) K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) K T (cid:3) trace (cid:2) P ≤ ( q ,...,q k ) (cid:3) = lim q →∞ · · · lim q k →∞ trace (cid:2) K T K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) (cid:3) trace (cid:2) P ≤ ( q ,...,q k ) (cid:3) = lim q →∞ · · · lim q k →∞ rank (cid:2) K T K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) = lim q →∞ · · · lim q k →∞ rank (cid:2) K ∗ T ( P ≤ ( q ,...,q k ) ⊗ I ) K T (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) = χ ( T ) . The proof is complete. (cid:3)
Theorem 3.4. If M is a graded invariant subspace of the tensor product F ( H n ) ⊗ · · · ⊗ F ( H n k ) , then lim q →∞ · · · lim q k →∞ rank (cid:2) P M ⊥ P ≤ ( q ,...,q k ) (cid:3) rank (cid:2) P ≤ ( q ,...,q k ) (cid:3) = lim q →∞ · · · lim q k →∞ trace (cid:2) P M ⊥ P ≤ ( q ,...,q k ) (cid:3) trace (cid:2) P ≤ ( q ,...,q k ) (cid:3) . This is equivalent to χ ( P M ⊥ S | M ⊥ ) = curv ( P M ⊥ S | M ⊥ ) . Proof.
Combining Theorem 2.4 with Theorem 3.3, and using Theorem 3.1 from [24], the result follows. (cid:3)
A closer look at the proof of Theorem 3.3, reveals that if T ∈ B n ( H ) is a pure element of finite rankand ∆ T ( I ) / T ∗ k,α k · · · T ∗ ,α k T ,β · · · T k,β k ∆ T ( I ) / = 0for any α , β ∈ F + n , . . . , α k , β k ∈ F + n k , except when | β | = | α | , . . . , | β k | = | α k | , thencurv ( T ) = χ ( T ) . Our final result is a version of the Gauss-Bonnet-Chern theorem for graded pure elements with finiterank in the noncommutative polyball.
Theorem 3.5.
Let T ∈ B n ( H ) , with n i ≥ , be a pure element of finite rank which is graded with respectto a gauge group U , and let H = M s ∈ Z k H s be the corresponding orthogonal decomposition. Then the spectral subspaces H s of U are finite dimensionaland there exist c , d ∈ Z k with c ≤ d such that H = L s ∈ Z k , s ≥ c H s and H := M s ∈ Z k , s ≥ d H s is an invariant subspace of T with the property that T | H is a finite rank element in B n ( H ) and curv ( T | H ) = χ ( T | H ) . Proof.
Assume that T is graded with respect to the gauge group U ( λ , . . . , λ k ) = X ( s ,...,s k ) ∈ Z k λ s · · · λ s k k Q ( s ,...,s k ) , ( λ . . . , λ k ) ∈ T k . The first part of the proof is similar to that of Theorem 3.3. Therefore, we can obtain that ∆ T ( I ) / Q ( s ,...,s k ) = Q ( s ,...,s k ) ∆ T ( I ) / , ( s , . . . , s k ) ∈ Z k . Since ∆ T ( I ) / is a positive finite rank operator in the commutatnt of { Q ( s ,...,s k ) } ( s ,...,s k ) ∈ Z k , we musthave that ∆ T ( I ) / Q ( s ,...,s k ) = 0 for all but finitely many ( s , . . . , s k ) ∈ Z k . Consequently, there are c = ( c , . . . , c k ) ≤ d = ( d , . . . , d k ) in Z k such that(3.11) ∆ T ( I ) / = X c ≤ s ≤ d ,...,c k ≤ s k ≤ d k D / s ,...,s k ) , ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 27 where D / s ,...,s k ) := ∆ T ( I ) / Q ( s ,...,s k ) . Taking into account that L s ≥ c ,...,s k ≥ c k H ( s ,...,s k ) is invariantunder each operator T i,j for any i ∈ { , . . . , k } and j ∈ { , . . . , n i } , we have(3.12) T ,β · · · T k,β k ∆ T ( I ) / H ⊆ M s ≥ c ,...,s k ≥ c k H ( s ,...,s k ) for any β i ∈ F + n i and i ∈ { , . . . , k } . Since T is a pure element in the polyball, the Berezin kernel K T isan isometry and(3.13) H = range K ∗ T = span { T ,β · · · T k,β k ∆ T ( I ) / h : β i ∈ F + n i , h ∈ H} . Hence, and using relation (3.12), we deduce that H = L s ≥ c ,...,s k ≥ c k H ( s ,...,s k ) . Due to relation (3.11)and the fact that ∆ T ( I ) / has finite rank, we have ∆ T ( I ) / H ⊆ M c ≤ s ≤ d ,...,c k ≤ s k ≤ d k M ( s ,...,s k ) ⊆ M c ≤ s ≤ d ,...,c k ≤ s k ≤ d k H ( s ,...,s k ) , where M ( s ,...,s k ) := ∆ T ( I ) / H ( s ,...,s k ) is finite dimensional. On the other hand, if β i ∈ F + n i with | β i | = m i ∈ Z + and i ∈ { , . . . , k } , we have T ,β · · · T k,β k ∆ T ( I ) / H ⊆ M c + m ≤ s ≤ d + m ,...,c k + m k ≤ s k ≤ d k + m k H ( s ,...,s k ) and T ,β · · · T k,β k ∆ T ( I ) / H is a finite dimensional subspace. Consequently, using relation (3.13) andthe fact that H = L s ≥ c ,...,s k ≥ c k H ( s ,...,s k ) , we conclude that all the spectral spaces H ( s ,...,s k ) are finitedimensional.Since T is a pure element in B n ( H ), we have I = ∞ X p =0 · · · ∞ X p k =0 Φ p T ◦ · · · ◦ Φ p k T k ( ∆ T ( I )) , where the convergence is in the weak operator topology. Hence, after multiplying to the left by Q ( q ,...,q k ) ,we obtain(3.14) Q ( q ,...,q k ) = ∞ X p =0 · · · ∞ X p k =0 Q ( q ,...,q k ) Φ p T ◦ · · · ◦ Φ p k T k ( ∆ T ( I ))for any ( q , . . . , q k ) ∈ Z k . On the other hand, taking into account relation (3.11), we deduce that(3.15) Q ( q ,...,q k ) Φ p T ◦ · · · ◦ Φ p k T k ( ∆ T ( I )) = d X s = c · · · d k X s k = c k Q ( q ,...,q k ) Φ p T ◦ · · · ◦ Φ p k T k ( D ( s ,...,s k ) )for any ( p , . . . , p k ) ∈ Z k + . Since ∆ T ( I ) ≤ I , it is clear that D ( s ,...,s k ) ≤ Q ( s ,...,s k ) . Taking into accountthat T i,j Q ( s ,...,s k ) = Q ( s ,...,s i − ,s i +1 ,s i +1 ,...,s k ) T i,j for any ( s , . . . , s k ) ∈ Z k , i ∈ { , . . . , k } , and j ∈ { , . . . , n i } , we haveΦ T i ( Q ( s ,...,s k ) ) = n i X j =1 T i,j Q ( s ,...,s k ) T ∗ i,j = Q ( s ,...,s i − ,s i +1 ,s i +1 ,...,s k ) Φ T i ( I ) Q ( s ,...,s i − ,s i +1 ,s i +1 ,...,s k ) ≤ Q ( s ,...,s i − ,s i +1 ,s i +1 ,...,s k ) for any i ∈ { , . . . , k } . Consequently, and using the inequality D ( s ,...,s k ) ≤ Q ( s ,...,s k ) , we deduce that(3.16) Φ p T ◦ · · · ◦ Φ p k T k ( D ( s ,...,s k ) ) ≤ Φ p T ◦ · · · ◦ Φ p k T k ( Q ( s ,...,s k ) ) ≤ Q ( s + p ,...,s k + p k )8 GELU POPESCU for any ( p , . . . , p k ) ∈ Z k + . Now, let ( q , . . . , q k ) ∈ Z k be such that ( q , . . . , q k ) ≥ ( d , . . . , d k ), Since { Q ( s ,...,s k ) } are orthogonal projections, relation (3.16) implies ∞ X p =0 · · · ∞ X p k =0 d X s = c · · · d k X s k = c k Q ( q ,...,q k ) Φ p T ◦ · · · ◦ Φ p k T k ( D ( s ,...,s k ) )= d X s = c · · · d k X s k = c k Φ q − s T ◦ · · · ◦ Φ q k − s k T k ( D ( s ,...,s k ) )= Φ q − d T ◦ · · · ◦ Φ q k − d k T k d X s = c · · · d k X s k = c k Φ d − s T ◦ · · · ◦ Φ d k − s k T k ( D ( s ,...,s k ) ) ! . Due to relations (3.14) and (3.15), we obtain(3.17) Q ( q ,...,q k ) = Φ q − d T ◦ · · · ◦ Φ q k − d k T k ( B ) , where B := d X s = c · · · d k X s k = c k Φ d − s T ◦ · · · ◦ Φ d k − s k T k ( D ( s ,...,s k ) ) . Due to relation (3.17), we deduce thatΦ T i (cid:0) Q ( q ,...,q k ) (cid:1) = Q ( q ,...,q i − ,q i +1 ,q i +1 ,...,q k ) for any ( q , . . . , q k ) ≥ ( d , . . . , d k ) and i ∈ { , . . . , k } . Consider the subspace H := M s ≥ d ,...,s k ≥ d k Q ( s ,...,s k ) H and let T | H := ( T | H , . . . T k | H ) with T i | H := ( T i, | H , . . . , T i,n i | H ) for i ∈ { , . . . , k } . An inductiveargument shows that, for any ( m , . . . , m k ) ∈ Z k + ,( id − Φ m +1 T | H ) ◦ · · · ◦ ( id − Φ m k +1 T k | H )( I H )= ( id − Φ m +1 T | H ) ◦ · · · ◦ ( id − Φ m k +1 T k | H ) X d ≤ s ≤ d + m ∞ X s = d · · · X s k = d k Q ( s ,s ,...,s k ) | H = · · · = ( id − Φ m k +1 T k | H ) X d ≤ s ≤ d + m · · · X d k − ≤ s k − ≤ d k − + m k − X s k = d k Q ( s ,s ,...,s k − ,s k ) | H = X d ≤ s ≤ d + m · · · X d k ≤ s k ≤ d k + m k Q ( s ,...,s k ) | H . In particular, for m = · · · = m k = 1, we have( id − Φ T | H ) ◦ · · · ◦ ( id − Φ T k | H )( I H ) = Q ( d ,d ,...,d k ) | H ≥ , which shows that T | H ∈ B n ( H ) and has finite rank. On the other hand, due to the relations above, theoperator ( id − Φ m +1 T | H ) ◦ · · · ◦ ( id − Φ m k +1 T k | H )( I H ) is an orthogonal projection for any ( m , . . . , m k ) ∈ Z k + .Therefore,trace (cid:2) ( id − Φ m +1 T ) ◦ · · · ◦ ( id − Φ m k +1 T k )( I ) (cid:3)Q ki =1 (1 + n i + · · · + n m i i ) = rank (cid:2) ( id − Φ m +1 T ) ◦ · · · ◦ ( id − Φ m k +1 T k )( I ) (cid:3)Q ki =1 (1 + n i + · · · + n m i i ) . Consequently, applying Theorem 1.2 and using [24] (see Theorem 1.3 and Corollary 1.4), we deduce thatcurv ( T | H ) = χ ( T | H ) . This completes the proof. (cid:3)
ULER CHARACTERISTIC ON NONCOMMUTATIVE POLYBALLS 29
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Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA
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